Loss of Prestress, Camber, and Deflection of Non-Composite ...publications.iowa.gov/21548/1/IADOT_HR_137_Loss... · of Noncomposite and Composite Structures Using Different ... cretes

Department of

College of Engineering

The University of Iowa

Iowa City, Iowa

Loss of Prestress, Camber, and Deflection

of Noncomposite and Composite Structures

Using Different Weight Concretes

Final Report

by

D. E. Branson

B.L.Meyers

K. M. Kripanarayanan

Report No. 70-6

Prepared Under Iowa State Highway Commission Research Project HR-137

August 1970

LOSS OF PRESTRESS, CAMBER, AND DEFLECTION

OF NON-COMPOSITE AND COMPOSITE STRUCTURES

USING DIFFERENT WEIGHT CONCRETES

Final Report

by

D, E. Branson Professor of Civil Engineering

B. L. Meyers Associate Professor of Civil Engineering

K. M. Kripanarayanan Research Associate in Civil Engineering

The opinions, findings, and conclusions expressed in this publication are those of the al\thors and not necessarily those of the Iowa State Highway Commission.

Report No. 70-6 Prepared Under Iowa State Highway

Commission Research Project HR-137

Department of Civil Engineering University of Iowa

Iowa City

August 1970

FOREWORD

This is a report of research conducted under the Iowa State

Highway Commission Research Project No. HR-137. The project was

initiated in February 1968. A progress report, No. 69-1, was sub

mitted in February 1969.

This project is being coordinated with the Iowa State Highway

Commission Research Project No. HR-136, "Creep and Shrinkage

Properties of Lightweight Concrete Used in the State of Iowa" (see

final report dated August 1970); and with the Iowa Highway Research

Board Project No. HR-104, ''Field Observation of Five Lightweight

Aggregate Pretensioned Prestressed Concrete Bridge Beams" (see

final report by J. A. Young).

Acknowledgment is made of the assistance of Messrs. S. E.

Roberts, Research Engineer, C. Pestotnik, Bridge Engineer,

Y. H. Gee, Assistant Bridge Engineer, and J. A. Young, Research

Technician, of the Iowa Highway Commission; and Mr. J. H. Boehm

ler, Jr., President, Pres tressed Concrete of Iowa, Inc.

The authors would also like to thank the Idealite Co., Denver,

Colorado, and the Hydraulic Press Brick Co., Brooklyn, Indiana for

donating materials used in the experimental program.

ii

ABSTRACT

Presented in this report are the results of an investigation of

the use of lightweight concretes in prestressed and reinforced con

crete structures. Both "sand-lightweight" and "all-lightweight" con

cretes are included in the study. The sand-lightweight concrete

cons is ts of 100% sand subs ti tu ti on for fines, along with Idea lite coarse

and medium lightweight aggregate and Type I cement. The all-light-

weight concrete consists of Haydite coarse, medium, and fine

aggregates along with Type I cement.

The study is divided into three parts: a materials study of

the concretes themselves, a laboratory study of the behavior of both

non-composite and composite beams that included prestressed (15

beams) and reinforced (3 beams) beams, and the field measurement

of camber of prestressed girders (5 girders) used in the fabrication

of a composite bridge in Iowa. The minimum test period for the

laboratory beams is 6 months, although data is recorded for 1 year

for 3 of the beams. The test period for the bridge girders is 560

days.

The laboratory prestressed concrete beams are designed in

five groups (3 beams in each group) to investigate the loss of

iii

pres tress, initial and time -dependent camber, load -deflection behavior

(under single and repeated load cycles) and the effect of different slab

casting schedules. One group of 3 reinforced beams is used to inves ti -

gate the initial and time-dependent deflection, load-deflection behavior

after sustained loading, and the effect of different slab casting sched-

ules.

The methods described for predicting material behavior and

structural response are generalized to apply to prestressed and rein-

forced structures of normal weight, sand-lightweight, and all-light-

weight concrete. Continuous time functions are provided for all needed

parameters, so that the general equations readily lend themselves to

computer solution. Approximate equations are also included,

Design procedures are presented for the following:

1. Calculation of strength and elastic properties, creep and shrinkage of the lightweight concretes of this project at any time, including ultimate values. An indication is also given of the calcula<· tion of these properties for other concretes in general.

2. Calculation of loss of prestress and camber at any time, including ultimate values, of non-composite and composite prestressed structures.

3. Calculation of deflections at any time, including ulti-mate values, of non-composite and composite reinforced structures.

4. Calculation of deflections of prestressed concrete mem-bers under single and repeated load cycles (with constant as well as increasing stress range). Calculation of deflections of reinforced concrete members under sustained loads in the non-linear range for short times (24 hours) is also included.

iv

Results computed by these methods are shown to be in good

agreement with the control specimen data, the laboratory beam data,

and the bridge girder data.

Published experimental data concerning the time-dependent

(prestress loss, camber, and deflection) effects and load deflection

response of prestressed and reinforced beams are shown to be in

reasonable agreement with the results computed by the design methods

presented in this report. Ranges of variation are also shown. These

data include normal weight, sand-lightweight and all-lightweight con-

crete, non-composite and composite members, and both laboratory

specimens and actual structures.

This project is thought to be the first such comprehensive study

of the initial plus time-dependent material behavior and related struc-

tural response of both non-composite and composite structures using

different weight concretes. A new procedure is also developed for pre-

dieting the entire load-deflection curve of both reinforced and prestressed

members under repeated load cycles into the cracking range.

Keywords: all-lightweight concrete; beams (structural); bridge girders; camber; composite construction (concrete to concrete); creep (materials); deflection; lightweight concrete; loss of prestress; modulus of elasticity; normal weight concrete; precast concrete; prestressed concrete, repeated cycle; sand-lightweight concrete; shrinkage; single cycle; steel relaxation; strain; stress; structural design; sustained; test beams; time-dependent.

v

TABLE OF CONTENTS

Chapter

1.

2.

3.

4.

List of Tables

List of Figures

Notation

INTRODUCTION -

1. 1 Statement of the Problem

1. 2 Objectives and Scope

1. 3 Review of Literature

DESCRIPTION OF EXPERIMENTAL INVESTIGATION

2. 1 Materials and Test Specimens

2.2 Instrumentation and Test Data

STRENGTH AND ELASTIC PROPER TIES, CREEP AND SHRINKAGE

3. 1 Strength and Elastic Properties

3.2 Creep and Shrinkage

LOSS OF PRESTRESS AND CAMBER

4. 1 Relaxation Tests

4. 2 Computed Loss of Prestress, Camber and Deflection

4.3 Required Calculations and Summary of General Parameters

vi

Page

ix

xi

xvii

1

1

2

3

8

8

10

12

12

16

21

21

23

38

TABLE OF CONTENTS (Cont'd)

Chapter

4.4 Sample Calculations

4.5 Experimental Loss of Prestress, Camber and Deflection Results

4.6 Discussion of Experimental Results and Conclusions

4.7 Comparison of Computed and Measured Data Reported by Others

4. 8 Summary of Results Reported by Others and Conclusions

5. LOAD DEFLECTION STUDIES OF PRESTRESSED AND REINFORCED CONCRETE BEAMS

6.

5. 1

5.2

5. 3

5.4

5.5

5.6

General

Single Cycle Load Tests of Prestressed Members

Repeated Load Tests of Prestressed Members

Increasing Load Plus 24-Hour Sustained Load Tests

Results Reported by Others

Summary and Conclusions

SUMMARY AND CONCLUSIONS

LIST OF REFERENCES

APPENDIX A Specimen Details

APPENDIX B Creep and Shrinkage Variables

APPENDIX C Specimen Details for the Data in the Literature

vii

Page

41

44

62

70

89

92

92

93

107

127

132

148

155

161

Ap 1

Ap 12

Ap 20

TABLE OF CONTENTS (Cont'd)

Chapter

APPENDIX D Camber Equations for Common Prestress Profiles

APPENDIX E Photographs of Laboratory Specimens

APPENDIX F Computer Flow Charts and Typical Outputs

viii

Page

Ap 29

Ap 31

Ap 37

LIST OF TABLES

Table Page

1 EXPERIMENTAL AND COMPUTED LOSS OF PRE STRESS FOR LABORATORY BEAMS AND COMPUTED LOSS OF PRESTRESS FOR BRIDGE GIRDERS 54

2 MEASURED AND COMPUTED MIDSPAN CAMBER & DEFLECTION FOR LABORATORY BEAMS & BRIDGE GIRDERS 56

3 COMPUTED ULTIMATE LOSS OF PRESTRESS AT MIDSPAN, BY TERMS, FOR THE LABORATORY BEAMS AND BRIDGE GIRDERS, USING THE GENER -AL EQUATIONS (14) & (17) WITH EXPERIMENTAL PARAMETERS 58

4 COMPUTED ULTIMATE MIDSPAN CAMBER, BY TERMS, FOR THE LABORATORY BEAMS AND BRIDGE GIRDERS, USING THE GENERAL EQS. (15), (16), (18) & (20) WITH EXPERIMENTAL PARAMETERS 60

5 WORKING LOAD, COMPUTED AND OBSERVED VALUES OF ULTIMATE LOAD AS WELL AS VALUES OF WORST DISCREPANCY BETWEEN COMPUTED AND OBSERVED DEFLECTION CURVES 103

6 DETAILS OF REPEATED LOAD CYCLES AND DIS-CREPANCY IN THE OBSERVED AND COMPUTED VALUES OF MIDSPAN DEFLECTION FOR BEAMS OF GRPS C & E 124

7 DETAILS OF INCREASING LOAD PLUS 24-HR SUS-TAINED LOAD TESTS WITH REGARD TO WORKING LOADS, ULTIMATE LOADS AND DEFLECTIONS UNDER THESE LOADS 131

lX

LIST OF TABLES (Cont'd)

Table Page

Al DETAILS OF LABORATORY BEAMS (GRPS A, B, C) AND BRIDGE GIRDERS Ap 2

A2 DETAILS OF LABORATORY BEAMS (GRPS D, E AND F) Ap 4

A3 DETAILS OF CONCRETE MIXES AND MIXING PROCEDURE FOR LT-WT CONCRETES Ap 6

A4 CONCRETE PROPERTIES (GRPS A, B, C AND BRIDGE GIRDERS), TEMPERATURE AND HUMIDITY DATA Ap 7

A5 CONCRETE PROPER TIES (GRPS D, E, & F), TEMPERATURE AND HUMIDITY DATA Ap 9

Ab CONCRETE PROPERTIES OF LAB BEAMS AT "LOAD-DEF" STUDIES Ap 11

Cl PROPERTIES OF TEST BEAMS AT UNIVERSITY OF FLORIDA (~) Ap 21

CZ PROPERTIES OF TEST BEAMS AT UNIVERSITY OF ILLINOIS (24} Ap 22

C3 PROPERTIES OF. TEST BEAMS AT TEXAS A &M UNIVERSITY (27) Ap 23

C4 PROPERTIES OF TEST BEAMS AT UNIVERSITY OF MISSOURI (2_!} Ap 24

C5 DETAILS OF BEAMS REPORTED BY ABELES (2.i) Ap 25

Cb DETAILS OF BEAMS REPORTED BY WARAWARUK, SOZEN & SIESS (Q) Ap 26

C7 DETAILS OF BEAMS REPORTED BY SHAIKH AND BRANSON (49) Ap 27

cs DETAILS OF BEAMS REPORTED BY BURNS & SIESS (2!) Ap 28

x

Figure

1

2

3

4

5

6

7

8

9

10

11

12

LIST OF FIGURES

Laboratory beams and bridge girders

Concrete strength vs time curves for lab concretes (Gps B, C)

Creep coefficient vs time curves for lab concretes (Gps A, B, C)

Shrinkage vs time curves for lab concretes (Gps A, B, C)

Concrete strength vs time curves for lab concrete (Gps E, F)

Creep coefficient vs time curves for lab concrete (Gps D, E, F)

Shrinkage vs time curves for lab concretes (Gps D, E, F)

Results of steel relaxation tests

Determination of experimental loss of prestress

Computed and experimental loss of prestress of beams of Group A (three non-composite beams)

Computed and experimental loss of prestress of beams of Groups B and C (two non-composite and four composite beams)

Computed and experimental loss of prestress of beams of Groups D and E (two non-composite and four composite beams)

xi

Page

9

14

14

14

15

15

15

22

22

45

46

47

Figure

13

14

15

16

17

18

19

20

21

22

23

24

LIST OF FIGURES (Cont'd)

Computed loss of prestress of five composite bridge girders

Computed and experimental midspan camber of beams of Group A (three non-composite beams)

Computed and experimental midspan camber of beams of Groups Band C (two non-composite and four composite beams)

Computed and experimental midspan camber of beams of Groups D and E (two non-composite and four composite beams)

Computed and experimental midspan deflection of beams of Group F (one non-composite and two composite beams)

Computed and experimental midspan camber of five composite bridge girders

Computed and experimental loss of prestress at end of beams reported in Reference (23)

Computed and experimental loss of pres tress at center of beams reported in Reference (~)

Computed and experimental midspan camber of beams reported in Reference (~)

Computed and experimental loss of prestress at center of beams reported in Reference (24)

Computed and experimental midspan camber of beams reported in Reference (24)

Computed and experimental loss of prestress at end of beams reported in Reference (2 7)

xii

Page

48

49

50

51

52

53

72

73

74

77

78

81

Figure

25

26

27

28

29

30

31

32

33

34

35

36

37


Computed and experimental loss of prestress at center of beams reported in Reference (2 7)

Computed and experimental midspan camber of beams reported in Reference (27)

Computed and experimental loss of prestress at end of beam reported in Reference (1.!_)

Computed and experimental loss of prestress at center of beam reported in Reference (31)

Computed and experimental midspan camber of beam reported in Reference (1.!_)

Two point loading for 'load-deflection' studies of laboratory beams

Moment of inertia of cracked section (Icr)

Observed and computed midspan deflection versus load curves for beams of Group A (three non-composite prestressed beams)

Observed and computed midspan deflection versus load curves for beams of Group B (one non-composite and two composite prestressed beams)

Observed and computed midspan deflection versus load curves for beams of Group D (three non-composite prestressed beams)

Details of deflections under repeated loadings

Sample calculations

Observed and computed midspan deflection vs load curve of beam Cl under 3 cycles of repeated loading {one non-composite prestressed beam)

xiii

Page

82

83

86

87

88

94

98

100

101

102

110

114

118

Figure

38

39

40

41

42


Observed and computed midspan deflection versus load curve of beam CZ under 3 cycles of repeated loading (one non-composite prestressed beam)

Observed and computed midspan deflection versus load curve of beam C3 under 3 cycles of repeated loading (one composite prestressed beam)

Observed and computed midspan deflection versus load curve of beam E 1 under 3 cycles of repeated loading (one non-composite prestressed beam)

Observed and computed midspan deflection versus load curves for beams E2 and E3 under 3 cycles of repeated loading (two composite prestressed beams)

Effect of repeated loading (in the cracked range) on total deflections of laboratory beams of Groups C and E

43 Observed and computed values of midspan deflection for beams of Group F under 24-hr sustained loading (one non-composite and two composite reinforced beams)

44

45

46

Observed and computed midspan deflection (using Eqs, (38) and (41) for beams under static loading as in (A) (Data from Reference 56) and as in (B) (Data from Reference 41)

Observed and computed midspan deflection (using Eqs, (38), (40), and (41) for beams under static loading as in (A) (Data from Reference 49) and for beams under repeated loading as in (B) (Data from Reference 54)

Comparison of computed and observed values of midspan deflection of beams in Reference (54), under two cycles of repeated loading (three non-composite reinforced beams)

xiv

Page

119

120

121

122

123

130

134

138

141


Figure Page

47 Range of validity of Eqs. (38), (40) and (41) for rectangular beams with different steel percentages - -included in this dimensionless plot are also the results from studies made on rectangular prestressed beams from Reference (!!_), (49), (54) and (56) as well as the current study 143

48

Bl

B2

B3

Range of validity of Eqs. (38), (40), and (41) for T beams with different steel percentages --included in this dimensionless plot are the results from the current study only

Strain components

Time -dependent strain variables

Nominal creep and shrinkage correction factors for the parameters shown from Ref. 18

E 1 View of laboratory showing beams in foreground and pre-

144

Ap 14

Ap 16

Ap 18

stressing bed containing additional beams at right Ap 32

E2 Forms for beams in prestressing bed Ap 32

E3 Strain gage indicator and switching and balancing unit used with load cells to measure pres tress force

E4 Prestressing bed, jacking equipment and beams stored

Ap 33

in bed Ap 33

ES Close-up of jacking equipment, bulkheads, and grips Ap 34

E6 Shrinkage specimens in foreground and 7 beams ( 1 beam crosswise in foreground). Two additional beams in prestressing bed Ap 34

E7 Two of 4 composite beams. Strain gage points and dial gages can be seen. Strands used in relaxation tests are seen at right Ap 35

xv


Figure

E8 Cylinders loaded in creep racks and Whittemore gage used to measure strains of beams and shrinkage and creep specimens

E9 View of beam C 1 showing the crack pattern prior to failure

ElO View of beam Cl after failure

xvi

Page

Ap 35

Ap 36

Ap 36

NOTATION

l = subscript denoting cast-in-place slab of composite beam or effect of slab

2 = subscript denoting precast beam

A = area of section

A = area of gross section, neglecting the steel g

As = area of tension steel in reinforced members and area of prestressed steel in prestressed members

I

As = area of compression steel in reinforced members and area of non-tensioned steel in prestressed members

At = area of transformed section

a = distance from end of beam to the nearest of 2 symmetrical disphrams. Also used as the distance from end to harped pt. in 2-pt. harping. Also used as empirical constant-see Eq. (1). Also used as distance of load from the near support--see Eq. (41).

b = empirical constant determined in the laboratory--see Eq. (1). Also used as distance between applied loads--see Eq. (41). Also used as compression flange width.

Cs = creep coefficient defined as ratio of creep strain to initial strain at slab casting.

= creep coefficient at any time t

= creep coefficient of the composite beam under slab dead load

= creep coefficient of the precast beam concrete

xvii

c u

C. F.

c

= ultimate creep coefficient defined as ratio of ultimate creep strain to initial strain

= correction factor to account for conditions other than standard

= subscript denoting composite section. Also used to denote concrete, as Ee

cp = subscript denoting creep

D = differential shrinkage strain. Also used as a subscript to denote dead load

DS = subscript denoting differential shrinkage

d = effective depth of section

E = modulus of elasticity

E = modulus of elasticity of concrete such as at 28 days c

E . = modulus of elasticity of concrete at the time of initial Cl

loading, such as at transfer of prestress, etc.

Ecs = modulus of elasticity of concrete at the time of slab casting

Es = modulus of elasticity of steel

e = eccentricity of steel

ec = eccentricity of steel at center of beam. Also used, as indicated, to denote eccentricity of steel in composite section

e0

= eccentricity of steel at end of beam

F = prestress force after losses

Fi = initial tensioning force

F0

= prestress force at transfer (after elastic loss)

xviii

f:,F

f:,F s

= loss of prestress due to time-dependent effects only (such as creep, shrinkage, steel relaxation). The elastic loss is deducted from the tensioning force, Fi' to obtain F

0

= total loss of prestres s at slab casting minus the initial elastic loss that occurred at the time of prestressing

= total loss of prestress at any time minus the initial elastic loss

= total ultimate loss of prestress minus the initial elastic loss

fc = concrete stress at steel c. g. s due to pres tress and pre-cast beam dead load

fed = concrete stress at steel c. g. s due to differential shrinkage

fcs = concrete stress at steel c. g. s due to slab dead load (plus

f ' Sl

f y

H

diaphragm dead load where applicable)

= compressive strength of concrete

= compressive strength of concrete at time t

= compressive strength of concrete at 28 days

= ultimate (in time) compressive strength of concrete

= modulus of rupture of concrete

= tensile strength of concrete

= stress in prestressing steel at transfer (after elastic loss)

= initial or tensioning stress in prestressing steel

= yield strength of steel (defined herein as O. 1% offset)

= relative humidity in percent

= moment of inertia of slab

= moment of inertia of precast beam

xix

= moment of inertia of composite section with transformed slab. The slab is transformed into equivalent precast beam concrete by dividing the slab width by E /E

C2 c 1

= moment of inertia of cracked transformed section

= effective moment of inertia

= moment of inertia of gross section, neglecting the steel

= effective moment of inertia under repeated loads

= moment of inertia of transformed section, such as an uncracked pres tressed concrete section

i ::: subscript denoting initial value

K = deflection coefficient. For example, for beams of uniform section and uniformly loaded: Also for

Shrinkage cantilever beam, K = 1/4 ~ = 1/2

simple beam, K = 5/48 ~ = 1/8 hinged-fixed beam, K = 8/185 , ~ = 11/128

(one end continuous)

fixed-fixed beam, K = 1/32 ~ = 1/16 (both ends continuous)

K1

= deflection constant for the slab dead load

K 2 = deflection constant for the precast beam dead load

~ = deflection coefficient for warping due to shrinkage or

k

temperature change -- see K for values of~

= distance of neutral axis from compression flange -- see Eq. (39), also kr = 0,85 - 0.45(As' /As)•

kr = reduction factor to take into account the effect of compres -

=

sion steel, movement of neutral axis, and progressive cracking in reinforced flexural members; and effect of nontensioned steel in prestressed flexural members, see k for values of kr

2 2 2 1 + e /r , where r =lg/Ag

xx

L

LA

M

= span length in general and longer span for rectangular slabs. Also used as a subscript to denote live load

= subscript denoting loading age

= bending moment. bending moment, formly loaded:

When used as the numerical maximum for beams of uniform section and uni-

cantilever beam , simple beam ,

hinged-fixed beam (one end continuous), fixed-fixed beam (both ends continuous).

(-) M = q L2/z (+) M = q L2 /8 (-) M = q L2 /8 (-)M=qL2/12

= maximum bending moment under slab dead load for composite beams

M 2 = maximum bending moment under precast beam dead load

M 10 = bending moment between symmetrically place diaphrams

Ms, Di = bending moment due to slab or slab plus disphram, etc., dead load

Mer = cracking moment

Mmax = maximum moment under service loads

m = modular ratio of the precast beam concrete, EsfEcs' at the time of slab casting. Also used as subscript to indicate measured values

n = modular ratio, Es/Eci' at the time of loading, such as at release of prestress for prestressed concrete members. Also usually used as Es/Ee for reinforced members

P = applied transverse load for load-deflection studies

P = applied transverse load corresponding to the cracking er moment, Mer

= maximum value of applied repeated transverse load in a cycle

xxi

pult

PG

PG cp

PL

PL l cp

PL cp

PL el

PL u

p

p'

Q

= applied transverse load corresponding to the ultimate strength of the beam

= prestress gain in percent of initial tensioning stress or force

= pres tress gain due to creep under slab dead load at time t

= prestress gain due to differential shrinkage at time t

= elastic prestress gain at slab casting

= total pres tress loss in percent of initial tensioning s tress or force

= prestress loss due to creep prior to slab casting at time t

= prestress loss due to creep after slab casting at time t

= prestress loss due to creep at time t

= prestress loss due to elastic shortening

= prestress loss due to steel relaxation at time t

= prestress loss due to shrinkage of concrete at time t

= total prestress loss at any time t

= ultimate prestress loss

= steel percentage, As/bd for cracked members, and As/Ag for uncracked members. Also used in percent in shrinkage warping equations

= compressive steel percentage, A~/bd for cracked members, and A~/Ag for uncracked members. Also used in percent in shrinkage warping equations

= differential shrinkage force - D A1 E 1/3. The factor 3 provides for the gradual increase in the shrinkage force from day 1, and also approximates the creep and varying stiffness effects.

xx ii

q

r

s

= uniformly distributed load

= radius of gyration, r 2 = lg/ Ag

= subscript denoting time of slab casting. Also used to denote steel. Also used as subscript to indicate sustained

load

sh = subscript denoting shrinkage

t

t

= total depth or thickness of section. Also subscript to denote time-dependent

= time in general, time in hours in the steel relaxation equation, and time in days in other equations herein

tLA = age of concrete when loaded, in days

u = subscript denoting ultimate value

w = unit weight of concrete in pcf

x = subscript to indicate distance as measured from the end of the beam -- see Eq. (35)

Yes = distance from centroid of composite section to centroid of slab

Yt = distance from centroid of gross section to extreme fiber in tension

= ratio of creep coefficient at any time to ultimate creep

coefficient, c/ cu

= ratio of creep coefficient at the time of slab casting to C u

= creep correction factor for the precast beam concrete age

when loaded

i3s = creep correction factor for the precast beam concrete age when the slab is cast for composite beams

Y s = ratio of shrinkage at slab casting to shrinkage at ultimate (referred to 7-day initial reading)

xx iii

y' = ratio of shrinkage after slab casting to shrinkage at ulti-s

(referred to 7-day initial reading) mate

I::. = deflection or camber

I::.. = initial deflection, camber 1

( l::.i ) 1 = initial deflection under slab dead load

( I::. i) lD = initial deflection due to diaphram dead load

( l::.i J2 = initial deflection under precast beam dead load

( I::. i )D = initial dead load deflection

( l::.i )F = initial camber due to the initial pres tress force, F 0

0

I::. DS = differential shrinkage deflection

I::. L = live load deflection

I::. t = total camber, deflection, at any time

I::. u = ultimate camber, deflection

(€sh)t = shrinkage strain in inches/inch or cm/cm, etc., at time t

( € ) = ultimate shrinkage strain in inches/inch or cm/cm, etc. sh u

cp = curvature

cpsh = curvature due to shrinkage warping -- see Eq. (16)

= curvature due to shrinkage warping of precast beam up to slab casting -- see Eq. (20)

* 1

= load ratio for repeated load studies -- see Eq. (40)

xxiv

1

Chapter 1

INTRODUCTION

1. 1 Statement of the Problem

As a result of the increased use of structural lightweight con

crete for precast bridge girders along with normal weight concrete

deck slabs, a need exists for a better understanding of the factors,

primarily time-dependent, that affect prestress loss and camber (in

the case of prestressed girders) and deflection (in the case of rein

forced girders) in composite beams of these materials. Of particular

interest in this study is the behavior of sand-lightweight (100% sand

substitution for fines along with lightweight coarse aggregate) and all

lightweight prestressed structures in relation to normal weight pre

stressed structures, and the effect of the composite slab on the ulti

mate loss of prestress and camber. The effect of composite slabs

on the deflection of reinforced concrete members is also included in

this study.

In order to complete a comprehensive study of the initial plus

time-dependent deformational behavior of non-composite and com-

posite structures, the load-deflection response of reinforced and pre

stressed members under single cycle and repeated cycle shQrt-time load

tests (with constant and increasing load levels) into the cracking range are

2

also included in this study. Twenty-four hour sustained load tests into

the cracking range are also studied.

1. 2 Objectives and Scope

The principal objective of this investigation is to evaluate

experimentally the time-dependent behavior of sand-lightweight and

all-lightweight concrete beams (pres tressed and reinforced), includ

ing composite beams, in order to present practical design methods,

and to give an indication of their accuracy for predicting loss of

prestress and camber (in the case of prestressed beams) and deflec

tions (in the case of reinforced beams).

The study is divided into three parts: a materials study of

the concretes themselves, a laboratory study of the behavior of both

non-composite and composite beams that included prestressed (15

beams) and reinforced (3 beams) beams, and the field measurement

of camber of pres tressed girders (5 girders) used in the fabrication

of a compos.ite bridge in Iowa. The minimum test period for the

laboratory beams is 6 months, although data is recorded for 1 year

for 3 of the beams. The test period for the bridge girders is 560 ·

days.

The laboratory prestressed concrete beams are designed in

five groups (3 beams in each group) to investigate the loss of pre

stress, initial and time -dependent camber, load-deflection behavior

3

(under single and repeated load cycles) and the effect of different slab

casting schedules. One group of 3 reinforced beams is used to inves -

tigate the initial and time-dependent deflection, load-deflection behav

ior after sustained loading, and the effect of different slab casting

schedules.

Results computed by the methods described for predicting ma

terial behavior and structural response are shown to be in good agree

ment with the control specimen data, the laboratory beam data, the

bridge girder data, and other published experimental data. Continuous

time functions are provided for all needed parameters, so that the gen

eral equations readily lend themselves to computer solution. Approx

imate equations are also included.

1. 3 Review of Literature

Shrinkage of concrete is its contraction due to drying and

chemical change. Various empirical equations are presented in the

literature ill• @l_, ill for predicting shrinkage strains. AC! Com

mittee 435 (.!1 has given a quantitative resume of available informa

tion on creep and shrinkage as applied to deflections of reinforced

concrete beams.

Concrete undergoes time-dependent deformations under the

action of sustained loads that are attributed to creep of the concrete.

The contributions of Lorman (21, McHenry (!1, Neville L?2_,

Ross @)_, and, Troxell, et al ffi are noted. Lorman and Ross

4

suggested the use of hyperbolic expressions for predicting creep

(used in this report in modified form). McHenry's concept of

"superposition technique for creep" is used in this report; for example,

in the case of creep under slab dead load. Neville's study of the

physical nature of creep is noted.

A number of creep theories and mechanisms of creep have

been reviewed by Neville (1), Ali and Kessler (_!_Q). and Meyers, et al

(!..!). Meyers and Neville (_g_) and Pauw and Chai (_!l) have summa

rized the primary factors that influence creep. The influence of the

size and shape of the member on creep and shrinkage was also

reported by Hansen and Mattock (14).

The principal articles referred to in this report on the subject

of creep and shrinkage of all-lightweight and sand-lightweight con

crete are those of Jones, et al (15), ACI Committee 213 (~),

Pfeifer (!1)• Christiason (!~),Schumann (!..2)• and this project(~).

Although the behavior of non-composite and composite pre -

stressed beams of normal weight concrete has been studied in

References (20) through (34), etc., (most of these referred to non-

composite beams only), it appears that no such investigation has

been made of composite pres tressed members of lightweight concrete.

Lofroos and Ozell (~.U were apparently the first to report

experimental results of time-dependent camber of prestressed con

crete beams. The specimens were two pairs of post-tensioned

5

normal weight non-composite beams under different prestress levels.

Branson and Ozell (23) examined experimentally the initial

plus time-dependent camber of both composite and non-composite

post-tensioned beams of normal weight concrete. Methods for cal·

culating camber were developed using certain experimentally deter

mined coefficients. The predicted results were in fair agreement

with the measured values. It was also concluded that camber tends

to reach an ultimate value relatively early compared to creep and

shrinkage, because of the offsetting effects of loss of prestress and

camber growth due to creep.

Corley, Sozen and Siess (24) discussed at great length the

reduced modulus method, the rate of creep method, and the super

position method in a study of the time-dependent camber of pre

stressed concrete beams. The rate of creep method was deemed

preferable on account of its relative simplicity. It was concluded

that time-dependent camber could be objectionably high, if there was

high stress gradient in the beam.

Sinno (27) in his study of lightweight non-composite pres tressed

bridge girders, concluded that hyperbolic functions can be used to

predict loss of prestress and camber (used in modified form in this

report). He also observed that camber tends to reach an ultimate

value relatively early as compared to creep and shrinkage.

6

Yang (28) in a recent study of lightweight non-composite pre-

stressed beams, concluded that creep under constant stress and

variable stress was proportional to the applied stress within limits

of about 40% of the ultimate strength.,

Methods used in this study for predicting loss of prestress

and camber were based in part on the papers of ACI Committee

435 (29)and Branson (.£2.), (~Q).

With respect to short-time deflection of prestressed members

under static and repeated loading, the works of Abeles (35) - (38}, - -Burns (39), Hutton (40), and Warawaruk, Sozen, and Siess ('.!.!._) are

noted. Abeles 1 work primarily deals with partially pres tressed mem-

bers under static and fatigue loading. In general, it is concluded that

maximum tensile stress of the order of the modulus of rupture of the

concrete may be permitted under working loads without any detrimen-

tal effects on the serviceability and safety of the prestressed

members.

Burns (21_) has presented a detailed analytical method for

obtaining the moment-curvature relationship for partially prestressed

beams. The study was limited to pres tressed concrete beams with-

out non-tensioned steel.

Warawaruk, et al (41) in a comprehensive study of noncom-

pas ite prestressed beams presented methods for the prediction of

deflections of prestressed members at the various loading stages.

7

This method, however, is too elaborate as a design procedure.

The procedure developed by Branson (.?.Q_), (j), (lQ_), (42)

for predicting the deflection of reinforced beams under single-cycle

loading and adopted for the 1971 ACI Code (~), and applied to pre

stressed beams by Shaikh and Branson (49), is extended in this study

to the prediction of deflections of both reinforced and prestressed

beams under repeated load cycles into the cracking range.

8

Chapter 2

DESCRIPTION OF EXPERIMENTAL INVESTIGATION

2. 1 Materials and Test Specimens

The details of the laboratory beams and bridge girders are

shown in Figure 1 and Tables Al and A2. The laboratory beams were

designed as follows:

Group A -- 3 non-composite beams with different prestress moments made of sand-lightweight concrete.

Group B -- 3 beams, two of which are composite beams. The beams are made of sand-lightweight concrete. The slabs (of normal weight concrete) were cast at 4 weeks and 10 weeks after the casting of the beams. The same prestress moment is used for the three beams.

Group C - - Same as Group B but with a different pres tress moment.

Group D -- Same as Group A but made of all-lightweight concrete.

Group E

Group F

Same as Group B but with a higher stress level.

3 reinforced (non-prestressed) beams, two of which are composite beams. The beams are made of sand-lightweight concrete. The slabs (of normal weight concrete) were cast at 4 weeks and 10 weeks after the casting of the reinforced beams. The same steel percentage is used for the three beams.

9

-3/4"TDT or 611 srr

t;::==.:=.:::::=:=c=.g~·=s=.==:::.:-==:===:;::! 1-eeej_ j.611.j

15 1

·-- c ~~-- s. ---

"" ii.. 15 1

, ··-

~ ~

(See Tables Al & AZ)

z1r or 3rr

_,::1: · 1CJ~~ J_ +

~6".]

Laboratory non-composite and composite beams

c .•. c. .. .. -··· c.g.s. . 14. 3 11

---~ "' .-.-34. 5' 17'

86'

16'

"""-"~8" djaphrams @ 29' from

1 2 supports 153

----·

4. 34. 5'-

Bridge girders

' t 6. 211

"'

_Lf-15"-j

3@2" = l" 3" - ~"

5@2 11

......._ __ ,_ 3-1/2

25"

~ 45"

oOOo~J 00099000 8 11 00000000

I !-1@2"J I ~19" --l

O Straight strands

• Deflected strands

Figure 1 Laboratory beams and bridge girders

10

The beams for groups A, B, C, D, E, and F were moist

cured for 3 days. Pres tressing was done at age 7 -9 days for the

beams of groups A, B, C, D and E. The reinforced beams of group

F were in position at age 21 days. The bridge girders (steam cured

until prestressed at age 2-3 days) are sand-lightweight concrete

( 100% sand substitution for fines along with lightweight coarse aggre -

gate), while the slabs are normal weight concrete. The composite

bridge deck was cast 9 weeks after the bridge girders were cast.

The concrete mix ingredients and the mixing procedure for

the different concretes are shown in Table A3. Two shrinkage speci

ments and 3 creep specimens (6" by 12" cylinders placed under a sus

tained uniform stress - see Tables A4 and A5) were cast for each

lightweight concrete.

2. 2 Instrumentation and Test Data

Steel collars with electrical strain gages (SR-4) mounted

thereon were used as load cells for individual strands to measure the

prestressing force applied to each laboratory beam.

Dial gages were used on both sides of each beam at midspan

to measure both initial and time-dependent camber of the laboratory

beams. A level rod and a precise level were used to obtain the

camber measurements for the bridge girders.

11

A Whittemore mechanical strain gage (10" gage length) was

used to measure the concrete strains of the creep and shrinkage

specimens and the laboratory beams.

The experimental data for the laboratory specimens consists

of the following:

1. Concrete strength properties, elastic properties, creep and shrinkage data from control specimens. Steel properties.

2. Temperature and humidity data.

3. Steel relaxation data.

4. Initial and time-dependent concrete beam strains. These are used in determining the experimental loss of pres tress.

5. Initial and time-dependent camber,

6. Load-deflection, cracking, and ultimate strength data.

Camber data for the bridge girders is included in this report

from Reference (32).

The concrete properties, temperature, and humidity data are

shown in Tables A4 and A5.

12

Chapter 3

STRENGTH AND ELASTIC PROPER TIES, CREEP AND SHRINKAGE

3. 1 Strength and Elastic Properties

A study of concrete strength versus time in this project and

Reference (18) indicates an appropriate general equation in the form

of Eq. (1) for predicting compressive strength at any time.

t (f~ l2sd ( 1 )

a+ bt

where a and bare constants, (£~)28d = 28-day strength, and tis time.

The following equations were developed in this study and

Reference (18), and used in Reference (33 ), for normal weight, sand-~ ~

lightweight, and.all-lightweight concrete (using both moist and steam

cured concrete, and types I and III cement). Eqs. (2) and (4) refer

to the concrete (type I cement) of this project:

Moist cured concrete, type I cement

13

Moist cured cone rete, type III cement

Steam cured concrete, type I cement

I t (fc)t = 1. 00 + O. 95t ( 4)

Steam cured concrete, type III cement

(5)

I

where tis age of concrete in days, and (fclu refers to an ultimate (in

time) value. The results of Eqs. (2) and (4) agree with the experi-

mental data of this project, as shown in Figures 2 and 5. As shown

in References(_~) and (42), Eqs. (2) - (5) refer to average values

only. See these references for ranges of variation.

The secant, initial tangent, and computed (using Eq. 6)

modulii of elasticity for the laboratory beams and bridge girder

concretes are shown in Tables A4 and A5.

E 1,5 ~ . I

c = 33w -.] fc , psi; win pcf and fc in psi ( 6)

The computed values for the limited number of tests were from 6%

- u ~

14

• (2) • • ---(4)

J~)_Moistcured, Calc--Eq (2)

HJ.Steam cured, Calc--Eq (4)

-6- Moist cured, Meas--Gp B

0 · 3 w---r---1f---1 •Moist cured, Meas - -Gp C

•Steam cured, Meas -Bridge o._ _ _. __ ...._ __ _.._ __ .._ __ .._ _ _. __ ...._ __ ~ 0 10 20 30 40 50 60 70

Age of concrete in days Figure 2. Concrete strength vs time curves for lab concretes (Gps B, C

and bridge girder concrete)

... u

l.6

11TTl~DF=f==rll ---

Measured

• Group A

A Group B

• Group C

-- - Computed by Eq. (7) o..__~ __ ...._ __ ..._ __ .._ __ .._ _ __. __ ...._ __ _.._ __ _._ _ __, 0 20 40 60 80 100 120 140 160 180 200

Time in days Figure 3. Creep coefficient vs time curves for lab concretes (Ops A, B, C)

Time in days Figure 4. Shrinkage vs time curves for lab concretes (Gps A, B, C)

1. 2

'"do. 9 00 N

-do. 6

" ....!:' - u ~ 0. 3

0

0

J f7

!'....

fl Eq.

10

15

.... -Moist cured,

_(.?l_ Cale - - Eq. (2) (2) -

Moist cured, _..__ Meas. -- Gp E

Moist cured,

--;- Meas. -- Gp F j I "I

20 30 40 50 60 70 80 Age of concrete in days

Fig. 5 Concrete strength vs time curves for lab concrete (Gps E, F)

1.6~~~~~--r---~~c-~--~~--~----,c-~--r-~~--~~c-~-.

... .:: " ..... u .....

::::: " 0 u P<

" " I-< u

----Measured

Iii Group D & Group E ct Group F

Computed by Eq. (7)

OL-----'-----"'------'-----.l......---'------L---.....1. ____ ..J.... __ --1 __ ---1

0 20 40 60 80 100 120 140 160 180 200 Time in days

Fig. 6 Creep coefficient vs time curves for lab concrete (Gps D, E, F)

600~~~~~~~~~~--~~--r---~~c-~--~~--~~--~~ Measured .::"' ..... I

.:: 0

·;;; -400 !:: >: Ol ..cl

~-S200

Gp D ---Gp E

Iii Group D & Group E ct Group

~==;!ct=1-qr-+---r--t--1L-#. Gp F

ell" ..!< oi Computed .:: " ..... ..c: by Eq. (8) ,. u o~--......J----__J_----...L----..L.----L---......J----__J_----~:.._-..:~.:.....:.-

r]J .s 0 20 40 60 80 100 120 140 160 180 200 Time in days

Fig. 7 Shrinkage vs time curves for lab concretes (Gps D, E, F)

16

to 15% higher than the initial tangent values. However, the computed

initial camber of the laboratory beams and bridge girders was in

agreement with the measured results (Table 4 }. Eq. 6, developed

in Reference (18), is considered satisfactory for normal weight,

sand-lightweight, and all-lightweight concrete.

3. 2 Creep and Shrinkage

The principal variables that affect creep and shrinkage are

outlined and discussed in Appendix B. The design approach pre-

sented herein for predicting creep and shrinkage refers to "standard

conditions" and correction factors for other than standard conditions.

Based largely on the data and information from References

and this project, the following design procedure (developed in this

project and Reference (.!..§.), and used in Reference (42 )}, is recom-

mended for predicting a creep coefficient and unrestrained shrinkage

at any time, including ultimate values. The general values suggested

for Cu and (€shlu should be used only in the absence of specific

creep and shrinkage data for local aggregates and conditions. How-

ever, the "time-ratio" part (right-hand side except for C and u

(€sh>ul of Eqs. (7) - (9) have been found (_!1} to apply quite generally.

As shown in References (.!..§.}and (42}, these general values of Cu

and (€sh>u refer to average values only. See these references for

ranges of variation.

17

Standard creep equation -- 3" or less slump, 40% ambient relative humidity, minimum thickness of member 6" or less, loading age 7 days for moist cured and 1-3 days for steam cured concretes

c u lO+t0.60

(7)

For the laboratory beam lightweight concretes (moist cured) of this project, the following values apply:

GrouE Load. Age

A, B, c 7 days D 7 E 9 F 21

Rel. Hum.

40% 50 50 50

Cu

1. 75 1. 87 1. 80 1. 63

For the bridge girder sand-lightweight concrete project -- Cu= 2. 15 for H = 40%. H was 70%. H = 70%, Cu= 0. 80(2. 15) = 1. 72.

(steam cured) of this From Eq. (12) for

General value suggested for all weights of structural concrete (both moist and steam cured concrete, types I and III cement) -- Cu= 2. 35 for H = 40%. From Eq. (12) for H = 70%, Cu= O. 80(2. 35) = 1. 88.

Standard shrinkage equations - - 3" or less slump, 40% ambient relative humidity, minimum thickness of member 6 11 or less

Shrinkage at any time after age 7 days for moist cured concrete

(8)

For the laboratory beams lightweight concretes (moist cured) of this project, the following values apply:

GrouE Ini. Read. Age A, B, c 7 days

D 7 E 9 F 21

Rel. Hum. 40% 50 50 50

( e:sh )u

650 x 10- 6 in/in. 540 510 385

18

General value suggested for all weights of structural concrete (both types I and III cement) -- (e:shlu = 800 x 10-6 in/in for H = 40%. From Eq. (13) for H = 70%, (€shlu = O. 70(800 x 10-6) = 560 x lo-6 in/in.

Shrinkage at any time after age l -3 days for steam cured concrete

For the bridge girder sand-lightweight concrete of this project - -(e:shlu = 560 x io-6 in/in for H = 40%. H was 70%. From Eq. (13) for H = 70%, (e:shlu = O. 70 (560 x 10-6) = 392 x 10-6 in/in.

(9)

General value suggested for all weights of structural concrete (both types I and Ill cement) -- (e: 8 h) = 730 x 10-6 in/in for H = 40%. From Eq. (13) for H = 70%, (e:~h)u = O. 70(730 x lo-6) = 510 x 10-6

in/in.

In Eqs. (7), (8) and (9), tis time in days after loading for

creep and time after initial shrinkage is considered.

Values from the Standard Eqs. (7) - (9) of Ct/cu and

(e:sh)t/(e:shlu are:

1 mth 3 mths 6 mths .!.L!:. 5 yrs

Ctf Cu, Eq. (7) 0.44 o. 60 0.69 0.78 0.90

(e:shlt/(e:sh)u, Eq. (8) 0.46 0.72 0.84 o. 91 o. 98

(€sh lt/ (€sh lu' Eq. (9) 0.35 0.62 0.77 0.87 o. 97

The lower creep and shrinkage for the concrete of this pro-

ject, as compared to the average or general values, was probably

due to the high concrete strengths attained. The computed (in Eqs. 7

and 8) and measured creep and shrinkage for the moist cured con-

crete of this project are shown in Figures 3, 4, 6 and 7.

19

Correction factors

All correction factors are applied to ultimate values. However, since creep and shrinkage for any period in Eqs. (7), (8), and (9) are linear functions of the ultimate values, the correction factors in this procedure may be applied to short-term creep and shrinkage as well.

For slumps greater than 3", see Figure B3.

For loading ages later than 7 days for moist cured concrete and later than 1-3 days for steam cured concrete, use

Eqs. (10) and (11) for the creep correction factors (_!_§).

Creep (C.F. )LA= l.25t~~ 118 for moist cured concrete (10}

Creep (C.F.)LA =I. l3t~~ 095 for steam cured concrete (II)

where tLA is the loading age in days. For example,

When tLA=lO days, mo. 20 30 60 90

cu. (C. F. )LA=O. 95, 0.87 o. 83 0.77 0.74

st. cu. (C. F. )LA =0. 90. 0.85 0.82 0.76 0.74

For shrinkage considered from other than 7 days for moist cured concrete and other than 1-3 days for steam cured concrete, determine the differential in Eqs. (8) and (9) for

any period starting after this time. For shrinkage of moist cured concrete from 1 day (used to estimate differential shrinkage in composite beams, for example}, use Shrinkage C. F. = 1. 20.

For greater than 40% ambient relative humidity, use Eqs. (12) and (13) for the creep and shrinkage correction factors (~), (43), (44 ).

Creep (C. F. }H = 1. 27 - O. 0067 H, H = 40%

Shrinkage (C. F. )H = 1. 40 - O. 010 H, 40% = H = 80% = 3. 00 O. 030 H, 80% = H = 100%

(12)

( 13)

20

where His relative humidity in percent. For example,

When H = 40"/o, 50 60 70 80 90

100

Creep (C. F. )H = 1. 00, 0.94 0.87 o. 80 0.73 D.67 0.60

Shrinkage (C. F. )H = 1. DO. 0.90 0,80 o. 70 0.60 D.30 o.oo

For minimum thickness of members greater than 6 11, see

Figure B3 for the creep and shrinkage correction factors, as a func -tion of length of drying and loading periods. For most design purposes, this effect (as shown in Appendix B) can be neglected for creep of members up to about 10" to 12" minimum thickness, and for shrinkage of members up to about 8" to 9" minimum thickness.

This method of treating the effect of member size was based

on information from References (14), (18), (44), and this project. - - -For large-thickness members, refer to the method of Reference (14),

and others, for relating size and shape effects for creep and shrink-

age to the volume/surface ratio of the members, etc.

Other correction factors for creep and shrinkage, which are usually not excessive and tend to offset each other, are described in Appendix B. For design purposes, these may normally be neglected.

21

Chapter 4

LOSS OF PRESTRESS AND CAMBER

4, 1 Relaxation Tests

Relaxation measurements were made for three different dia-

meter 7-wire pres tressing strands. The results agreed well with the

equation suggested in Reference (45), as can be seen in Figure 8.

It should be noted, however, that the relaxation of steel stress

in a prestressed member takes place under decreasing steel strain

(due to creep, shrinkage, etc.), rather than at constant length as in a

relaxation test. The loss of prestress due to steel relaxation is also

affected by slab casting (level of stress in steel is raised) in the case

of composite beams. Due to these effects and the practice of over

tensioning to counteract the relaxation that takes place between the

time of tensioning and effective bonding of concrete to steel (this

practice was assimilated in the laboratory beam tests, where it is

noted in Figure 8 that about 2% relaxation takes place in 24 hours,

for example), it is felt that about 75% of the steel relaxation in a

constant-length relaxation test should be used in prestressed concrete

loss calculations.

i:: ·~ ..... i:: nl

·~ 0 ..... ·~ ·~ ..... i:: nl ·~

>< .... " nl 0 " ..... a.> a.> ..... ... ... i:: .....

a.> " ..... u a.> ... a.> a.> ..... p.. [fJ

l5

10

0

22

f . Log10

t

10 - 0, 55)100, ~1

= 0, 73 for tests

10

--· Experimental results of this project -----

100

-- ----r

Computed using above equation

[::, 1/4" a 3/8" 0 5/16"

1, 000 10, 000

Time in hours !log scale)

dia dia dia

100,000

Figure 8. Results of steel relaxation tests

1, 5"

I\ I I

Time afte pres tress

Top G_a,,_g_e __

Dotted line is computed initial

value 5.5" \ I I Bot.

o\ ing in day_,__ _ _., l"

400 800 400 800 1200

End Section Midspan Section

Initial plus time-dependent strain distribution diagrams from concrete strains measured on the sides of the beams

Typical experimental pres tress loss determined for end section at 180 days

fsi = 172 ksi, Es = 27 x 103 ksi, Observed cone. strain at cgs =

1001 x 10-6 in/in, Loss from meas. strains= (1001x10-6)(27x10 3 )(100)/172 = Inc. in meas. loss due to laterial dis tr. (det. as 2. 5% of 15. 7p Meas. loss due to steel relaxation (75% of value from Figure 8) = Total experimental loss of prestress

15. 7% o .. 4 5.5

21. 6%

Figure 9. Determination of experimental loss of prestress

23

It was concluded in Reference (46) that steel relaxation is

probably insignificant beyond 100, 000 hours (11, 4 yrs), and that this

ultimate value might be taken as twice the value at 1000 hours (1. 4

mths ). The relaxation equation recommended in this paper is the

same time-function (Log t) as that of Reference (45 ), except reduced

by 25% in magnitude and incorporating the idea of Reference (46) that

the ultimate value be taken as twice the value at 1000 hours. This

results in an ultimate steel relaxation for pres tressed concrete of

7. 5%, as shown in Term (4) of Eq. (14). Although Term (4) of

Eq. (14) was suggested on the basis of relaxation studies of 7-wire

prestressing strands used for pretensioned specimens, it is felt that

this is valid even for post-tensioned specimens (see comparison of

loss of prestress and camber of other published data in Sec. 4. 7).

4, 2 Computed Loss of Pres tress, Camber, and

Deflection(~), (24), (e), (~), (1.Q_), (2l_), (42), (45), (46), (47)

Non-composite beams at any time, including ultimate values

The loss of prestress, in percent of initial tensioning stress,

is given by Eq. (14).

( 1) (2) (3) ( 4)

(14)

where:

24

Term (1) is the prestress loss due to elastic shortening=

F· F·e2 Mne 1 1 fc = - + -- -- and n is the modular ratio at the time

At It It

of prestressing. Frequently F0

, Ag' and lg are used instead of Fi'

At, and It' where F 0

= Fi (1 - n p). Only the first two terms for fc

apply at beam ends.

Term (2) is the prestress loss due to concrete creep. The

AFt expression, Ct(l -2F ), was used in References (23) and (30/ to

0

approximate the creep effect resulting from the variable stress his -

tory. See the section on Required Calculations and Summary of

General Parameters for approximate values of AF /F (in form of t 0

fJ. F /F and fJ. F /F ) for this secondary effect at various s 0 u 0

times.

Term (3) is the prestress loss due to shrinkage (47). The

express ion, ( €shlt Es, somewhat (approximately 1 % loss differential

for the bridge girder ultimate value in the example herein) overesti-

mates (on safe side) Term (3).

Term (4) is the prestress loss due to steel relaxation,

Assumes Max. value= 7. 5% (at or above 105 hrs = 11. 4 yrs). In

this term, tis time after initial stressing in hours. This expression

applies only when f 8 /fy is greater than or equal to O. 55, in which fy

is the O. 1%-offset yield strength.

25

The camber is given by Eq. (15). It is suggested that an

average of the end and midspan loss be used for straight tendons

(laboratory beams herein) and 1-pt. harping, and the midspan loss

for 2-pt. harping (bridge girders herein).

where:

Term (1) is the initial camber due to the initial prestress

force after elastic loss, F0

• See Appendix D for common cases of

prestress moment diagrams with formulas for computing camber,

Here F = F. (1 - n f /f .), where f is determined as in o i c s1 c

Term (1) of Eq. (14).

Term (2) is the initial dead load deflection of the beam.

(lli)D = K M L 2 /Eci lg. See Notation for K and M formulas.

Term (3) is the creep (time -dependent) camber of the beam

due to the prestress force. This expression includes the effects of

creep and loss of prestress; that is, the creep effect under variable

stress. llFt refers to the total loss at any time minus the elastic

loss. It is noted that the term, tiFt/F0

, refers to the steel stress

or force after elastic loss, and the prestress loss in percent, PL

(as used herein), refers to the initial tensioning stress or force.

26

f . 6 Ft The. two are related as:

Fo PL)~

el f , and can be

bFt closely approximated by F =

0

0

Term (4) is the dead load creep deflection of the beam.

Term (5) is the live load deflection of the beam.

The deflection at any time for a non-prestressed reinforced

beam is given by Eq. ( 16 ).

where:

bt

(2) (3) ~

2 = - (6i)D - kr Ct (l>i)D - Kw <Psh L

Term (1) is the in.itial dead load deflection of the beam.

2 (6· )D = K M L /E . I • See Notation for K and M formulas.

1 Cl g

Term (2) is the dead load creep deflection of the beam. kr

takes into account the movement of the neutral axis. See Notation

for values of kr•

Term (3) is the deflection due to shrinkage warping.

( 16)

(6sh )f~<Psh L 2 See Notation for values of Kw; cp sh =. 7 (E:sh )tp l/

3 /t

where pis the steel percentage and tis the thickness of the member.

Term (4) is the live load deflection of the beam.

Unshored and shored composite beams at any time, including

ultimate values

Subscripts 1 and 2 are used to refer to the slab (or effect of

27

the slab such as under slab dead load) and precast beam, respectively.

The loss of prestress, in percent of initial tensioning stress,

for unshared and shored composite beams is given by Eq. (17),

(8)

~]100 DS f .

Sl

where:

(n f )(Ct c 2

(3)

6Fs+6Ft 12 - c )(1 - )-

s2 2F0 Ic

( 1 7)

Term (1) is the prestress loss due to elastic shortening.

See Term (1) of Eq. (14) for the calculation off . c

Term (2) is the prestress loss due to concrete creep up to the

time of slab casting. C is the creep coefficient of the precast beam S2

concrete at the time of slab casting. See Term (2) of Eq. (14) for

6 Fs comments concerning the reduction factor, ( 1 -

2 F ).

0

Term (3) is the pres tress loss due to concrete creep for any

period following slab casting. C is the creep coefficient of the t2

precast beam concrete at any time after slab casting. The reduction

factor, (1 - t:.Fs + 6Ft), with the incremental creep coefficient, 2 F 0

28

(Ct2

- Cs2

), estimates the effect of creep under the variable pre-

stress force that occurs after slab casting. The reduction factor

term was modified from previous references. The expression,

12/Ic• modifies the initial value and accounts for the effect of the

composite section in res training additional creep curvature (strain)

after slab casting.

Term (4) is the prestress loss due to shrinkage. See Term

(3) of Eq. (14).

Term (5) is the prestress loss due to steel relaxation. In

this term tis time after initial stressing in hours. See Term (4)

of Eq. (14) for the maximum value and limitations.

Term (6) is the elastic prestress gain due to slab dead load,

and mis the modular ratio at the time of slab casting.

MS, Die

Ig , Ms

0. refers to slab or slab plus diaphram dead

' 1

load, and e, Ig refer to the precast beam section properties for

unshared construction and the composite beam section properties

for shored construction.

Term (7) is the prestress gain due to creep under slab dead

load. Ctr is the creep coefficient for the slab loading, where the

age of the precast beam concrete at the time of slab casting is

considered.

29

Term (8) is the prestress gain due to differential shrinkage.

QYcsec PGDS = mfcd' where fed = Ic , and fed is the concrete stress

at the steel c. g. s. See Notation for additional descriptions of terms.

Since this effect results in a prestress gain, not loss, and is normally

small (see Table 3), it may usually be neglected.

The camber of unshored and shored composite beams is given

by Eqs. (18) and (19), respectively.

Unshored construction:

(1) (2)

( 5)

[-

{IF s + ( 1

Fo

(4)

+ (1 -

(6)

I2 - Cs ({11.)2 - (C - C ) ({I.)

2 t 2 S 2 l 2 IC

( 8) (9) ( 10)

where:

(3)

Term (1) is the initial camber due to the initial prestress

force after elastic loss, F • See Appendix D for common cases of 0

( 18)

30

prestress moment diagrams with formulas for computing camber,

( t,1.)F . See Term (1) of Eq. (15) for determining F .

0 0

Term (2) is the initial dead load deflection of the precast

beam. See Notation for Kand M formulas.

Term (3) is the creep (time-dependent) camber of the beam,

due to the prestress force, up to the time of slab casting. See Term

(3) of Eq. (15) and Terms (2) and (3) of Eq. (16) for further explana-

tion.

Term (4) is the creep camber of the composite beam, due to

the prestress force, for any period following slab casting. Again,

see Term (3) of Eq. (15) and Terms (2) and (3) of Eq. (16) for further

explanation.

Term (5) is the creep deflection of the precast beam up to the

time of slab casting due to the precast beam dead load.

Term (6) is the creep deflection of the composite beam for any

period following slab casting due to the precast beam dead load.

Term (7) is the initial deflection of the precast beam under

slab dead load. ( t. .) 1 = K M 1 L 2 /E I . See Notation for Kand 1 cs g

M formulas. When diaphrams are used, add to ( t.i)l:

L2 a2 ( 8 - 6), where M 1D is the moment between dia-

phrams, and a is L/4, L/3, etc., for 2 symmetrical diaphrams at

31

the quarter points, third points, etc., respectively.

Term (8) is the creep deflection of the composite beam due to

slab dead load. Ctl is the creep coefficient for the slab loading,

where the age of the precast beam concrete at the time of slab cast-

ing is considered.

Term (9) is the deflection due to differential shrinkage. For

simple spans, t;DS = Qy L 2/8E I, where Q = D A 1 E 1/3. See cs cs c

Notation for additional descriptions of terms. The factor 3 provides

for the gradual increase in the shrinkage force from day 1, and also

approximates the creep and varying stiffness effects @2_). This factor

3 is also consistent with the data herein and elsewhere. See Table 4

for numerical values herein. In the case of continuous members,

differential shrinkage produces secondary moments ( similar to

effect of prestressing but opposite in sign--normally) that should be

included.

Term (10) is the live load deflection of the composite beam,

in which the gross -section flexural rigidity, E I , is normally used. c c

Shored construction:

t;t = Eq. (18), with Terms (7) and (8) modified as follows: ( 19)

Term (7) is the initial deflection of the composite beam under

slab dead load. ( t;i)l = K M 1 L2 /Ecs Ic. See Notation for Kand

M formulas.

32

Term (8) is the creep deflection of the composite beam under

slab dead load= Ct1

( lli)1

• The composite-section effect is already

included in Term (7).

The deflection of ordinary reinforced composite beams of

unshared and shored construction is given by Eqs. (20) and (21 ).

Unshared construction:

( 1 ) ,-----J'---,

(4)

(2)

(5)

k (Ct r 2

- K cp W SS

2 L 2 - K ( ) w 'Psh - 'Pss 2 2

(7) (8) ( 9)

- k r

(3)

12 - C )( ti. )D I

s2 1 c

Term ( 1) is the initial dead load deflection of the beam.

2 ( t.1.)0 = KM L /E .I • See Notation for Kand M formulas.

Cl g

Term (2) is the dead load creep deflection up to the time of

(20)

slab casting. k takes into account the movement of the neutral axis. r

See Notation for values of k r

Term (3) is the creep deflection of the composite beam for

any period following slab casting due to the precast beam dead load.

33

Term (4) is the deflection due to shrinkage warping up to the

time of slab casting. See Term (3) of Eq. (16) for further explanation.

Term (5) is the deflection due to shrinkage warping for any

period following slab casting due to the shrinkage of the precast beam.

See Term (3) of Eq. (16) for further explanation.

Term (6) is the initial deflection of the precast beam under

slab dead load. ([1.) 1 = KML2/E I. SeeNotationforKandM 1 cs g

formulas. When diaphragms are used, add to ([Ii) 1 :

L2 a2 (B - b ), where M 1D is the moment between dia-

phragms, and a is L/4, L/3, etc., for symmetrical diaphragms at

quarter points, third points, etc., respectively.

Term (7) is the creep deflection of the composite beam due to

slab dead load. Ct is the creep coefficient for slab loading, where l

the age of the precast beam concrete at the time of slab casting is

considered.

Term (8) is the deflection due to differential shrinkage. See

Term (9) of Eq. (18) for further explanation.

Term (9) is the live load deflection of the composite beam, in

which the gross-section flexural rigidity, E I , is normally used. c c

Shored construction:

At = Eq. (20), with Terms (6) and (7) modified as follows: (2 1)

34

Term (6) is the initial deflection of the composite beam under

slab dead load. ( A.)1

= KM L 2 /E I . See Notation for Kand M l cs c

formulas.

Term (7) is the creep deflection of the composite beam under

slab dead load = C (A .) 1• The composite-section effect is already t1 1

included in Term (6).

It is suggested that the 28-day modulii of elasticity for both

slab and precast beam concretes, and the gross I (neglecting the

steel), be used in computing the composite moment of inertia, I , c

in Eqs. (17), (18), (19), (20), and (21).

Special case of "ultimate loss of prestress, camber, and deflection

For computing ultimate values of loss of prestress and camber,

Eqs. (22) - (29) correspond term by term to Eqs. (14) - (21), respec-

tively.

PL u

=

Loss of prestress for non-composite beams, as per Eq. (14):

(2)

(n f )C ( 1 c u

(4) ,----A----,

+ o. 075 f . J Sl

AFu --)

2F 0

(3)

(22)

35

Camber of non-composite beams, as per Eq. ( 15):

( I) (2) (3)

~ ~

( 6F 6F

6 = (6 i)F - (6 i)D t - F u + (1 __ u)c) (6 i)F u 2 F u

0 0 0 0

(4) (5)

~ ,--A..._.,

- Cu (6 i)D - 6L (23)

Deflection of non-composite non-pres tressed reinforced beams,

as per Eq. (16):

(1) (2) (3) (4)

~ ~ ,-"-----,

6 = - (6 i)D k C (6 .)0

K L2 - 6L (24) u r u i w cpu

Loss of prestress for unshared and shored composite beams,

as per Eq. (17): ( 1 ) (2) ( 3)

PL u

= [ (n f c) + (n f }{et C )(I 6F 6F +6F 1

2 - 2 Fs )+ (nf )(1-et )C (1- s u) l c s u c s u 2 F

0 0 c

(4) (5)

+ (c h) E /(1 + npk ) + O. 075 f . - (m f ) S U S S Sl CS

(7) ( 8)

12 - (m f )( (3 C ) -

1 cs s u c

~

- PG ] .!Q£ DS f .

Sl

(25)

A u

36

Camber of unshared composite beams, as per Eq. (18):

(1) (2) (3)

= (A .)F 1 0

(-A; s + (1

0

(4)

AF +AF

AF s

- --) Ct 2 F s

0

(

. AF - AF + - u s

F + ( 1 - s u )( 1

2 F - et )C) (A. )F 0

(5) ~

-etC(A.)2 s u 1

(9) (10)

s u 1 0 0

(6) (8)

- (1 - et )C (A.)2 s u l

l c

(26)

Deflection of unshared composite non-prestressed reinforced

beams, as per Eq. (20):

( 1 ) (2)

r--"----.

A = - (A 1. )D - k a. C (A . ) -

u rsu 1D

(5)

2 -Ky cpL

w s 1 u

- A L

(6)

(3)

k (1 - et )C (A.JD r 1!I u 1

(7)

k ~ C (A.) I -r s u i

(4) ~

12 2 l

-K y cp L w s u

c (8)

,---'--..

12 - ADS l

c

(2 7)

37

Camber of shored composite beams, as per Eq. (19):

6u " Eq. (26), except that the composite moment of inertia is used

in Term (7) to compute (6i}1

, and the ratio I2/Ic' is eliminated in

Term (8). (28)

Deflection of shored composite non-prestressed reinforced

beams, as per Eq. (21):

6 = Eq. (27), except that the composite moment of inertia is used u

in Term (6) to compute { 6i)l' and the ratio I2 /Ic, is eliminated in

Term (7). (2 9)

It is noted that Eqs. (14) - (29) could be greatly shortened by

combining terms and substituting the approximate parameters given

below, but are presented in the form of separate terms in order to

show the separate effects or contributions to the behavior (such as

due to prestress force, dead load, creep, shrinkage, etc., that

occur both before and after slab casting.

Grossly approximate equations:

Non-composite beams (prestressed) --

6 =6.+6.C(l u 1 1 u

6. = ( t:,. )F 1 1 0

(30)

Composite beams (prestressed)

c (1 + 2u) - n f + (£ h) E + O. 075 fs 1·J cs s u s

(31)

= A 1• + A. C

1 u

12 (-I ),

c

38

Non-composite beams (non-prestressed)

2 = -(A i)D - Cu ( Ai)D - Kw (<Psh)u L ' where

<Pshu = Y 8

(c h) /t, and K . is defined in Notation. s u w

Composite beams (non-prestressed) --

12 2* A ~ Ai + Ai Cu (-

1 ) - K (<Psh)u L , where

u c w

K is defined in Notation.. w

4. 3 Required Calculations and Summary of General Parameters

(32)

{33)

(34)

Continuous time functions are provided for all needed material

parameters (and for different weight concretes, moist and steam

cured), so that the equations herein readily lend themselves to com-

puter solutions. Certain other read-in data (such as for the effect of

behavior before and after slab casting--a , ~ , m, and AF /F ) s s s 0

are also included. The parameters related to material properties are

summarized below, so that for composite beam hand calculations for

example; in addition to the section properties, prestress force, F , 0

and concrete stresses, fc• fcs' the only calculations needed for com-

puting pres tress loss and camber are the initial camber, deflections - -

'~ The ratio 12 /le is dropped out for the shrinkage term to account for the cumulative effects of shrinkage - i.e., before slab casting, after slab casting and due to differential shrinkage. For values of

y s' see Section 4. 3.

39

The following loss of prestress ratios at the time of slab

casting and ultimate are suggested for most calculations:

t::.F /F for 3 wks to 1 mth between prestressing and slab s 0 .

casting = 0, 11 for Nor. Wt., O. 13 for Sand-Lt. Wt., O. 15 for All-Lt. Wt.

t::.F /F for 2 to 3 mths between prestressing and slab s 0

casting = O. 15 for Nor. Wt., O. 18 for Sand-Lt. Wt., O. 21 for All-Lt. Wt.

f::.F /F = O. 22 for Nor. Wt., O. 25 for Sand-Lt. Wt., 0. 31 u 0

for All-Lt. Wt.

Note that these are defined as the total loss (at slab casting

and ultimate} minus the initial elastic loss divided by the prestress

force after elastic loss. The different values for the different weight

concretes are due primarily to different initial strains (because of

different E's} for normal stress levels.

I

The following average modular ratios are based on fc = 4000

to 4500 psi for both moist cured (M. C.} and steam cured (S. C.} con-

crete and type I cement; up to 3-mths f~ = 6360 to 7150 psi (using

Eq. 2) for moist cured and 3-mths fb = 6050 to 6800 psi (using Eq. 4)

for steam cured, and for both 250 Kand 270 K prestressing strands:

40

Modular Nor. Wt. Ratio (w = 145)

M.C. S.C. At release of prestress n == 7.3 7.3

For the time bet- = 3 weeks, m= 6. 1 6. 3 ween prestressing 1 month, 6.0 6.2 and slab casting: 2 months, 5.9 6. 1

3 months, 5.8 6.0

SandLt. Wt.

(w = 120) M.C. S.C. 9.8 9.8

8. 1 8.3 8.0 8.2 7.9 8.2 7. 7 8.0

Es = 27 x 106 psi for 250 K strands, Es = 28 x 10 6

All-Lt. Wt.

(w = 100) M. C. S. C. 12.9 12.9

10. 7 10. 9 10. 5 10.7 10.3 10.6 10.2 10.5

psi for

2 70 K strands, CL refers to the part of the total creep that takes place s

to. 60 before slab casting (CLs =

0 60 , as per Eq. 7), and 13 ( = the 10 + t • s

avg. Creep (C.F. )LA from Eqs. 10 and 11) is the creep correction

factor for the pre cast beam concrete age when the slab is cast (under

slab dead load). See Eqs. (7), (8), (9), and the correction factors

herein, for suggested values for C and ( e: h) • u s u

The following may be substituted for normal weight, sand-

lightweight, and all-lightweight concrete (moist and steam cured,

and types I and III cement):

For the time bet- = 3 weeks, CLS = o. 38, 13 s = 0.85 ween prestressing 1 month, o. 44, 0.83 and slab casting: 2 months, 0.54, 0.78

3 months, o. 60, 0.75

The following may be substituted for normal weight, sand-.

lightweight and all-lightweight concrete (moist cured) and Types I

and III cement for composite non-prestressed beams.

41

(For 'beam in position' at 7 days)':'

For the time bet-ween 1beam in pos -ition' and slab casting

4. 4 Sample Calculations

= 2 weeks, 3 weeks, 1 month, 2 months, 3 months,

y = o. 29, y s s 1 o. 38,

o. 46, o. 63, o. 72,

= 0. 71 0.62 0.54 0.37 o. 29

The following numerical substitutions for ultimate lass of

prestress at midspan, using Eqs. (17), (25), and ultimate midspan

camber, using Eqs. (18), (26), with the general parameters given

herein, are made for the sand-lightweight, steam cured composite

bridge girders (with slab moist cured) of this project:

Parameters and terms for interior girders

Span = 86 ft, girder spacing = 7 ft, 2-point harping at

O. 4L-pt. from end, e (midspan) = 14. 3 in, e (end) = 6. 2 in, f . :: Sl

190,000 psi, F. = 867 kips, A = 4.56 in2 , Ag= 520 in2 , p = 0.00883, 1 s

lg= 108, 500 in4 , MD (precast beam) = 410 ft-k, IC = 334, 100 in 4

(using slab width divided by a factor of E t /E 1 b = 3. 42/3. 41 = s em s a

1. 00), Ms, Di (slab plus diaphram moment at midspan) = 630 ft-k.

Modulii of elasticity (using Eqs. 2, 4, and 6 for concrete):

E = 28 x 106 psi, as suggested for 270 K grade strands herein. s

~:~ The differentials are to be used when the beam is 'in position 1 at an age other than 7 days. Eg: For a slab cast at age of beam = 35 days with the beam in position at age = 28 days, the values of Ys and Ys are (O. 46 for 35 days - 7 days = 1 month minus O. 38

1 -- --for 28 days - 7 days = 3 w.eeks) = O. 08, and ( 1. 00 - O. 46) = O. 54, respectively.

42

Slab Ec = 3,41 x 106

psi, for f~ = 3500 psi, w = 145 pcf (Table A4).

Precast beam -- (see description of m and n in general parameters

section herein for concrete properties),

E /n 6 2.86 x 106 psi, E = = 28 x 10 /9. 8 = ci s

E = E /m = 28 x 106 /8. 2 = 3, 42 x 106 psi.

cs s

Using Fi' At' and It, as per Term (1) of Eq. (14) or (17) or

(25) f = 2467 psi. As per Term (6) of Eq, (16) or (21), f = 1006 c cs

psi. These concrete stresses refer to the midspan section. As per

Term (1) of Eq. (15) or (18) or (26), for camber, F = F. (1 - n f / 0 l c

fsi) = 758 kips, using fc = 2467 psi.

From the general parameters section: n = E /E . = 9 8· S Cl • '

for 2 months period between prestressing and slab casting --

m = E /E = 8. 2, a, = 0, 54, \3 = O. 78, llFs/F = O. 18; llF /F = s cs s s 0 u 0

o. 25.

From Eqs. (7) and (9), for H = 70%, C = I. 88, (€ h) = u s u

510 x l0-6 in/in,

Initial camber and deflection, and differential shrinkage

deflection:

( lli)F = 4. 09 in, as per Term (1) of Eq. (15) or (18) or (26). 0

( lli)2 = I. 74 in, as per Term (2) of Eq, (15) or (18) or (26).

( lli)l = 2. 26 in, as per Term (7) of Eq. (18) or (26). This

deflection is due to the slab and diaphram dead load.

43

ADS = O. 49 in, as per Term (9) of Eq. (18) or (26).

Solutions for interior girders

Ultimate loss of prestress at midspan using Eq. (25):

(1) (2) (3) (4) (5) (6) (7) (8) PL = 12.7 + 11.7 + 2.8 + 6.5 + 7.5 - 4.3 - 2.0 - 1.6 = 33.3%

u

Ultimate midspan camber using Eq. (26) minus AL:

(1)

Au = 4. 09 -

(2) (3) (4) (5) (6) (7) (8) (9) 1. 74 + 3. 05 + o. 80 - 1. 77 - o. 48 - 2. 26 - 1. 06 - o. 49

::::: 0.14in.

Ultimate loss of prestress at midspan using the approximate

Eq. (31):

PL = 24.6 - 5.2 + 7.5 + 7.5 = 34.4%. u

Ultimate midspan camber using the approximate Eq. {32):

Au= 0.09+ 0.05 = 0.14 in, where Ai= 4.09 - 1.74 - 2.26 = 0.09 in.

Tabulated in Tables 1, 2, 3 and 4 are the prestress loss,

camber, and deflection results by the more reliable Eqs. ( 14) - (18 ),

(20) and (22) - (27), and the approximate Eqs. (30) - (34), for the

laboratory beams and bridge girders. Although the agreement above

is good {note the camber is near zero due to the slab effect for the

bridge girders) by these methods, the approximate method may be

suitable in many cases for rough calculations only (see Tables 1 - 2).

Also, the calculations needed by the approximate methods are not

significantly fewer than by other methods. The more reliable

44

equations should be preferable for computer use.

4. 5 Experimental Loss of Prestress, Camber, and Deflection

Results

The loss of pres tress at the end and midspan for the labora

tory beams was determined from the measured concrete strains.

However, this measured loss does not include the steel relaxation

loss, since steel relaxation is a "stress relaxation at ,constant length

--or nearly so in the case of a prestressed concrete beam" pheno

menon. Separate relaxation tests were made and the results shown

in Figure 8. From these and other tests, the relaxation equation

given in Term (4) of Eq. (14) was determined. An example of the

experimental determination of prestress loss for a typical laboratory

beam is shown in Figure 9.

Experimental and computed loss of prestress versus time

curves for the laboratory beams are shown in Figures 10, 11 and

12, and the computed curves for the bridge girders in Figure 13.

Measured and computed midspan camber versus time curves for the

beams and girders are shown in Figures 14 - 18. The general Eqs.

(14) - (18), (20) with experimental parameters were used in all com

parisons with test results in Figures 14 - 18. These results are

shown in Tables 1 - 4 at release of prestress (camber only), just

before slab casting (3 and 9 weeks for the beams and 9 weeks for

30

20

"' "' Q) 10 ... ..., "' -C1l

·r< 0 ...,

·r< '1

·r<

'H 30 0 ...,

'1 Q)

u ... 20 Q) p.

'1 ·r<

"' 10

"' Q) ... ..., 0 "' Q) ...

p. 'H 0 30

"' "' 0 -- 20 C1l ..., 0

£-< 10

0

45

e m - - ---- ----\_ --\: '-- - I- - - - - - - - - - - ---- - - - ~ - --- ~ .-- ~

~ c;..- "\..

Beam Al

e m --- -_ 'l.-~ 1.-:l-: - .. 1----· --· -- - -- - ~ - -;:; .... - v v

..-. p

Beam AZ

e m

\ __ \. -- t-==- ~-:_-:.. I- - -- -

----~ - ~----- -- - ---- ~ - ~ u

' ,,,. Beam A3

0 40 80 120 160 200 240 280 320 360 Time after prestressing in days

Experimental ....- End -e- Midspan

Computed __ e__ End

_JP __ Midspan

Figure 10 Computed and experimental loss of prestress of beams of Group A (three non-composite beams)

"' "' " ... .., "' ~ ro ..... .., ..... Q .....

'+< 0 .., Q

" u ... " p.

30

20

Q 10 ..... t/l t/l

" ... .., t/l

" ... p.

..... 0

0

30

20

10

0 0

46

m e e I m ----\ ----- ~ •

~ Beam Bl Beam Cl

' slab cast e \ (3 weeks) • ----- ~

slab cast

(3 we~e~k~s~)~;:=::!::;;:~~~~~~ e m

~ ~-,.--,.,,.. •

Beam B2 Beam CZ

' slab cast e m

(9 weeks) i I

slab cast e m (9 weeks) I I

- -. -~

,pP ~ - ~

Beam B3 Beam C3

40 80 120 160 200 0 40 80 120 160 200 Time after prestressing in days

Experimental

- End -e- Midspan

Computed

--~-End _ .!l.l _Mids pan

Figure 11 Computed and experimental loss of prestress of beams of Groups B and C (two non-composite and four composite beams)

"' ., " ... ..... ., -"' ..... ..... ..... '1 .....

4-< 0 ..... '1

" u ... " p..

'1 ..... ., ., " ... ..... "' " ... p..

4-< 0 ., "' 0 --"' ..... 0

f-<

60.0

40.0

20.0

0

60.0

40.0

20.0

0

60.0

40.0

20.0

0

47

e, m I

e -

t - • - - l -,.... _,,a ~

~

I

•

'

,,,, • 0

r Beam Dl

slab cast (3 weeks)

e e m I I

j - t -- - - - - --, Beam DZ

slab cast e m

(9 weeks) e

- -- - ~

r Beam D3

40 80 120 160 200 0 40 80 120 Time after prestressing in days

Experimental

---- End -- Midspan

Computed

--~-End __ 11}_ Midspan

m ' 1 -

Beam El

m

Beam E2

m

--

Beam E3

160 200

Figure 12 Computed and experimental loss of prestress of beams of Groups D and E (two non-composite and four composite beams)

"' "' 0 ~

'+< 0

30

20

10

0

30

20

10

0

30

20

10

0

-1 ---· ---- . __ ""' .... ,. __ .... ,~ ::.::.-=....-::. -=--......... I -. , .,

""' '"-- Exterior m e Girder 162

-~ -- ,_ ___ . ••

,~ 1::.-----..1 ~ .. --.. ---- - -I -

I I •

~ .,, Interior m e Girder 153

,1 -· I~ ---i----· i:=== I"=:' __ • .----

I -I ,

,!'- I"" Interior

e m Girder 154

0 75 150 225 300 375 450 525 Time after prestressing in days

·~· ~-= ....... _ ... ----- t- ___ .. _

"--------I I,

~," Interior e m

Girder 155

,'I --=----- Jr.:-:.-..: -=--------· -~

'x 1-_-_-___ __ _...,.:

-=~ :.--

I , , ,, I"" Exterior

e m Girder 156

0 75 150 225 300 375 450 525 Time after prestressing in days

Girders moved to site 7-8 weeks after prestressing

Slab cast 9 weeks after pres tressing

--~-End m Midspan

Figure 13 Computed loss of prestress of five composite bridge girders

o. 6

0.4 -

0.2

0

o. 6

0.4

0,2 r 0

0.6

0.4

-0,2 v-0

0

-- - -

40

-

-

-

49

---- - - --- -

Beam Al

-

Beam A2

- -

Beam A3

80 120 160 200 240 280 320 360 Time after pres tressing in days

Experimental Computed

Figure 14 Computed and experimental midspan camber of beams of Group A (three non-composite beams)

Q .... ... " ..0

8 "' u Q a Ill 't:l .... ;;E

. 6

• 4

• 2

0

• 6

.4

• 2

0

.6

• 4

• 2

0

50

- -------r

,. Beam Bl Beam Cl

slab cast slab cast (3 weeks) (3 weeks)

4.- - - - ---~ .J - - -

••• • --- - .

Beam B2 . Beam CZ

slab cast slab cast (9 weeks) (9 we!:~

" ~ .. ·-- - -, • -. -~

r ........ • - -

Beam B3 Beam C3

0 40 80 120 160 200 0 40 80 120 160 200 Time after prestressing in days

Computed • Experimental

Figure 15 Computed and experimental midspan camber of beams of Groups Band C (two non-composite and four composite beams)

5 1

1. 2

- - -

-0.8 ,. • . -- - -

0.4

Beam Dl Beam El

0

{/}

Q)

..<:: 1. 2 u Q .....

I

slab cast Q

0.8 ..... (3 weeks)

... Q)

..0

8 0.4 <d

I~ - -,_

u Q Beam DZ Beam EZ <d 0 ~ {/}

"' ..... :::8 1. 2 I

slab cast (9 weeks)

0.8 --/" ~

-...._ - --- - -0.4

Beam D3 Beam E3 0

0 40 80 120 160 200 0 40 80 120 160 200 Time after prestressing in days

- -- -- Computed • Experimental

Figure 16 Computed and experimental midspan camber of beams of Groups D and E (two non-composite and four composite beams)

"' Q)

.c u ~ ..... >1 ..... >1 0 ..... .... u Q) .....

"""' Q)

'O

~

'" p..

"' 'O ..... ~

52

Time after 'beam in position' in days

0 0

20 40

-0.2 ~- - --0.4

-o. 6

0

-0.2

~lab cast (1 week)

- - ----0.4

-0.6

0

-0.2 ~

slab cast (7 weeks) - .

l -0.4

-0.6

60 80

-•--

---

--- Computed

100 120 140 160 180

1..-_._ -- ·--.. .. '

Beam Fl

- -

Beam F2

-Beam F3

-- Experimental

Figure 17 Computed and experimental midspan deflection of beams of Group F (one non-composite and two composite beams)

3. 0 -r " Steel

2. 0 Being Placed Exterior

Girder 152

"' 1.

<lJ .c c ' - - - - --- --u c r .... c

3 • .... .... <lJ

o~ ,.0

s 2. ro 0 u Interior c ro P< 1.

Girder 153

"' 'O .... s ~ 0

• 4

0 • -ro +' 3. 0 E-<

-; ~·

2. Interior

Girder 154 1. ' ..... _. •• ( - -

0 75 150 225 300 375 450 525

Time after prestressing in days

_, r 1~

• Interior

Girder 155

' .-... ... _ • -- ~ - ~ - - -• • • ~:

Exterior Girder 156

4 ,_ --- ---· ~ --- --i - -- --- -

' ... --0 75 150 225 300 375 450 525

Time after prestressing in days

Girders moved to site 7-8 weeks after prestressing

Slab cast 9 weeks after prestressing

-- Measured ----Computed

Figure 18 Computed and experimental midspan camber of five composite bridge girders

-

•

Beam No.

Al AZ A3 Bl B2 B3 Cl CZ C3 DI DZ D3 El EZ E3

a TABLE 1

EXPERIMENTAL AND COMPUTED LOSS OF PRES TRESS FOR LABORATORY BEAMS AND COMPUTED LOSS OF PRESTRESS FOR BRIDGE GIRDERS

bT" Computed Loss by d .

1me Computed Ultimate Loss

Bet. Computed Experi- General Eqs. ( 14), Gen. Eqs. Ult. Eqs. Approx. Pres. Loss Just mental ( 17) with exp. pa ram. ( 14 ), ( 1 7) (ZZ ), (25) Eq. (31)

& Slab Before Loss at at 180d for Lab. B with exp. with gen. with gen.

Cast Slab Cast 180 days and 560d for bdg gird. pa ram. pa ram. oaram.

Mid Ratio End Mid End Ratio Mid Ratio End Mid End Mid IEnd Mid

Laboratorv Beams e - - - 23.5 Z2.0 Z5.5 1. 09 24.6 1. lZ 31. 7 30.5 36.9 35.4 - -- - - Z l. 0 19.5 Z3.Z 1. 10 2Z.3 1. 14 Z8.9 Z7.8 33.5 3Z. 1 - -- - - 19.0 18. 5 21. 4 1. 13 20.4 1. 10 26.7 Z5.5 32. 0 30.6 - -- - - Z l. 6 21. 0 Z4.0 1. 11 ZZ.9 1. 09 Z9.8 Z8.6 34.6 33. 1 - -

2ld 15. 0 1. 07 Z l. 9 20.5 zz.z 1. 02 zo. 7 1. 01 Z6.5 25.0 Z8.9 Z7.Z 31. 0 Z9.4 63d 19.4 1. 10 Z l. 4 zo. 0 2Z.6 1. 06 21. 1 1. 05 Z6.8 25.Z 29.4 Z7.6 31. 0 Z9. 4

- - - 25. 0 Z4.0 Z5.7 1. 03 24.7 1. 03 31. 9 30.8 37.2 35.7 - -Zld 16. 4 0.97 Z3. 0 Z l. 4 23.7 1. 03 Z2.4 1. 05 Z8.Z Z6.7 30.9 29. 3 33. 1 31.6

63d Z l. 1 1. 01 Z3.6 2Z. 3 Z4.4 1. 03 Z3.0 1. 03 28.7 Z7.2 31. 7 30.0 33. 1 31.6

- - - 36.Z 35. 0 36.9 1. oz 35.8 1. oz 45.6 44.2 53. 9 5Z. 1 - -- - - 33.0 31. 0 3Z. 3 0.98 31. 0 1. 00 40.0 38.5 46.9 44.9 - -- - - 31. 9 Z8.0 30.5 0.96 Z9.Z 1.04 37. 9 36. 3 44.8 43. 0 - -- - ·- 32. 0 Z9.0 31.2 0.98 30.2 1. 04 38.7 37.5 46.Z 44.8 - -

19d Z0.9 1. oz Z8. 0 Z5.0 27. 0 0.96 Z5.3 1. 01 31. 1 Z9.4 35.4 33.4 37.8 36.0

6ld Z6. 1 1. 00 30. 0 Z8.0 28.7 0.96 Z7.0 0.96 3Z.7 30.9 36.8 34.8 37.8 36.0

TABLE 1 (Cont'd)

bTime Computed Loss by ctcomputed Ultimate Loss

Bet. Computed Experi- General Eqs. ( 14), Gen. Eqs. Ult. Eqs. Approx. Beam Pres. Loss Just mental (17) with exp. param. ( 14 ), ( 1 7) (22 ), (25) Eq. (31) No. & Slab Before Loss at at 180d for Lab. B with exp. with gen. lwith gen.

Cast Slab Cast 180 days and 560d for bdg gird. par am. par am. pa ram.

Mid Ratio End Mid End Ratio Mid Ratio End Mid End Mid End Mid

Bride:e Girders 152 65d 28.4 - - - 27. 3 - 27.6 - 29.5 29. 9 30.4 34.0 30.5 35.0 153 65d 29.4 - - - 28. 1 - 28.0 - 30. 3 30. 1 30.3 33. 3 3 o. 5 34.4 154 65d 29.4 - - - 28. 0 - 28. 0 - 30.2 30. 1 30. 3 33. 3 30. 5 34.4 155 60d 28.4 - - - 27. 1 - 26.6 - 29. 3 28.7 30. 3 33. 3 30.5 34.4 156 60d 29. 8 - - - 28.3 - 28. 9 - 30.5 31. 0 30.4 34.0 30.5 35.0

a All losses are expressed in percent of initial stress. The ratios in the table are: Computed/ Experimental. See Footnote b, Table 3, for a description of experimental parameters.

b

c

d

e

The laboratory beams and bridge girders were prestressed at age 7-9 days and 2-3 days, respectively.

See Figure 9 for an example of the experimental loss determination. The 180 day and 560 day times in the table refer to times after prestressing.

The laboratory beam concrete strengths (for Gps. A-C) at release were well beyond the range specified for the general parameters; so then and m values for these lab. beams were computed separately. However, for the lab. beams of Gps. D and E, the suggested n and m values are used. Where general parameters are used, a correction factor is applied for rel. hum. only.

No approximate equation was given for non-composite beams for loss of prestress.

Beam No.

Al

A2 A3 Bl B2 B3 Cl C2 C3 Dl D2 D3 El E2 E3

eFl eF2 eF3

aTABLE 2

MEASURED AND COMPUTED MIDSPAN CAMBER&: DEFLECTION FOR LABORATORY BEAMS &: BRIDGE GIRDERS

bTime Comp. camber by d

Computed Ult. Camber Bet. Camber just Gen.Eqs. (15), (16) Gen. Eqs. Ult. Eqs. Approx.

Initial Camber Prest. Before (18) &: (20)withexp. (15), (16), (23), (24), Eqs. (30) &: Slab Cast param, @ 180d for (18), (20) (26), (27) (32), (33)

Slab lab. B &: 56 Od - Br.G. with exp. with gen. (34) with Meas Comp Ratio Cast Meas Comp Ratio Meas Comp Ratio par am. pa ram. gen. par.

Laboratory Beams 0.27 0.25 0.93 - - - - 0.44 0.46 1. 04 0.54 0.68 0.77 o. 2 0 o. 19 o. 95 - - - - o. 35 o. 35 I. 00 0.42 0.52 0.59

BadD o. 15 - - - - - 0.27 0.26 0.96 o. 31 0.38 0.44 0.22 0.22 I. 00 - - - - o. 39 0.39 I. 00 0.46 0.58 o.66 0.23 0.22 0.96 2ld 0.32 0.32 1. 00 0.25 0.27 I. 08 0.28 0.26 0.29 o. 23 0.22 o. 96 63d 0.36 0.35 0.97 0.26 0.27 I. 04 0.28 0.28 0.30 o. 2 7 o. 2 7 1. 00 - - - - 0.47 0.49 1. 04 o. 57 o. 73 0.75 o. 2 7 0.27 I. 00 2ld 0.39 0.39 I. 00 0.34 o. 36 1. 06 0.38 0.37 o. 39 o. 2 7 0.27 I. 00 63d 0.44 0.44 1. 00 0.35 o. 37 I. 06 0.39 0.39 o. 39 0.56 0.54 o. 96 - - - - 0.98 o. 95 0.97 1. 10 I. 44 1. 6 7 0.43 0.45 I. 05 - - - - 0.84 0.82 0.98 0.94 1. 19 I. 39 o. 41 0.40 0.98 - - - - 0.75 0.73 0.97 0.86 I. 05 1. 24 0.42 0.42 I. 00 - - - - 0.78 0.77 0.99 0.90 1. 12 1. 2 9 0.42 0.43 1. 02 19d 0.62 o. 59 o. 95 0.52 0.52 I. 00 0.55 0.58 o. 51 0.42 0.43 1. 02 6ld 0.72 o. 71 0.99 o. 54 0.57 1. 05 0.59 0.62 o. 51

- 0.07 - - - - - 0.34 0.30 0.89 0.38 0.47 0.55 - 0.07 - 7d o. 13 o. 14 I. 08 o. 32 0.28 0.88 o. 30 0.45 0.58 - 0.07 - 5ld 0.25 0.24 0.96 0.45 0.40 0.89 0.43 0.59 0.58

TABLE 2 (Cont'd)

bT. Comp. camber by d

Camber ime Computed Ult.

Bet. Camber Just Gen, Eqs. (lS), (16) Gen. Eqs. Ult. Eqs, Approx.

Beam Initial Camber Prest. Before (18) & (20)with exl\ (lS), (16), (23), (24), Eqs. (30),

No, & Slab Cast param, @ 180d for (18), (20) (26), (27) (32), (33)

Slab lab.B & S60d-Br,G, with exp. with gen. (34)with

Meas Comp Ratio Cast Meas Comp Ratio Meas Comp Ratio pa ram par am. gen.par.

Brid1 e Girders 1S2 2.0S 2, 14 1. 04 6Sd 3. 10 3,06 0,98 a.so 0.47 0,93 0,4S 0,Sl O,S3

1S3 2. OS 2.22 1. 08 6Sd 3. 10 3. 13 1. 02 0,2S 0.21 0.84 o. 17 o. 14 o. 14 1S4 2. 10 2.22 1. 06 6Sd 3,0S 3. 13 1, 03 o. 20 o. 21 1. OS o. 17 o. 14 o. 14 lSS 1.90 2, 14 1. 13 60d 2.9S 3,04 1. 03 -0. 02 0, 07 - o. 01 o. 14 o. 14 1S6 l,8S 2.27 1. 23 60d 2.92 3. 16 1. 08 o. 30 O,S4 cl, 80 a.so o. S l O,S3

a All camber values are in inches, Ratios are: Computed/Measured. See Footnote b, Table 3, for a description of experimental parameters. Also, see Sample Calculations for a description of general parameters.

b

c

d

e

See Footnote b, Table 1. Beams Fl-F3 were in position at beam age= 21 days.

Camber has Figure 18 ), is 0, 22".

been reduced from about 3" before slab casting to less than 1/2" after 1 year (see This ratio is large for the near zero camber, even though the difference in camber

See Footnote d, Table 1,

The camber of beams Fl, F2, and F3 being non-pres tressed reinforced beams are negative in magnitude, i, e., the values in this table for the beams (F 1, F2, F3) refer to deflections,

Beam No,

Al A2 A3 Bl B2 B3 Cl C2 C3 Dl D2 D3 El E2 E3

a, bTABLE 3

COMPUTED ULTIMATE LOSS OF PRESTRESS AT MIDSPAN, BY TERMS, FOR THE LABORATORY BEAMS AND BRIDGE GIRDERS,

USING THE GENERAL EQUATIONS (14) & (17) WITH EXPERIMENTAL PARAMETERS

Creep Creep El. Gain Creep Gain Due Total Loss, Elast, Loss Loss Shrink Relax Due to Gain Due to Diff. Eqs, ( 14). Loss Before After Loss Loss Slab to Slab Shrink (17)

Slab Cast Slab Cast .

Laboratorv Beams 5,2 8.0 - 9.8 7.5 - - - 30,5 4. 1 6.3 - 9.9 7,5 - - - 27.8 3.2 4,8 - 10, 0 7.5 - - - 25.5 4,5 6.9 - 9.7 7. 5 - - - 28,6 4.5 2.9 1,2 9. 7 7,5 -0.4 -0,2 -0.2 25.0 4,5 4.0 o. 9 9.7 7.5 -0.4 -0.2 -0.8 25.2 5.4 8,3 - 9.6 7,5 - - - 30.8 5,4 3,5 1, 5 9.6 7,5 -0,4 -0,2 -0.2 26.7 5,4 4,8 1. 1 9,6 7.5 -0.4 -0.2 -0.6 27.2

11,2 18,2 - 7,3 7.5 - - - 44.2 8. 9 14,6 - 7.5 7.5 - - - 38.5 8. 0 13,2 - 7.6 7,5 - - - 36.3 8. 8 14,0 - 7.2 7. 5 - - - 37,5

8. 9 5,6 1. 5 7.2 7,5 -0,7 -0,2 -0.4 29.4 8.9 8.2 1, 1 7. 1 7,5 -0.7 -0.2 -1. 0 30.9

<.n 00

a, bTABLE 3 (Cont'd)

Creep Creep El. Gain Creep Gain Due Total Loss,

Beam Elas t. Loss Loss Shrink Relax Due to Gain Due to Diff. Eqs. ( 14 ),

No. Loss Before After Loss Loss Slab To Slab Shrink (17)

Slab Cast Slab Cast

Bridge Girders 152 11. 5 9.8 2.2 4.6 7. 5 -3. 7 -1. 5 -o. 5 29. 9 153 12. 0 10. 3 2. 3 4.5 7. 5 -4.2 -1. 7 -o. 6 30. 1

154 12. 0 10. 3 2.3 4.5 7.5 -4.2 -1. 7 -o. 6 30. 1

155 11. 5 9.6 2.2 4.5 7. 5 -4.3 -1. 7 -0.6 28. 7 156 12. 3 1 o. 3 2.4 4.4 7.5 -3. 8 -1. 5 -0.6 31. 0

aThe table is arranged in order of terms in Eq. (17). All losses are expressed in percent of initial s tress.

bThe experimental parameters used in the calculations for this table are shown in Tables A4 and AS and elsewhere herein for the lightweight concretes of this project. The slab shrinkage is shown here only. The correction factors given herein for age of loading, humidity, and member thickness (8" for Br. Gir.) are used where appropriate with the experimental parameters. The resulting creep and shrinkage factors used are:

Avg. Rel. Humidity Precast Beam Creep Precast Beam Shrink

(x 10- 6 in/in)

Slab Shrink. (from day ~l used in comp. diff.str. (x 10 in/in)

c = u (e h) = s u

( e h) = s u

Laboratory Beams Gp. A, B, C Gp D ~

40% 50% 50% 1.75 l.87 1.80 650 540 510

_Qp_X 50%

1. 63 385

* * 470

(only for Gps. 440

B & C) 440

Bridge Girder 152-156

70% 1.62

352

330

Also see the Sample Calculations for a comparison with the general parameter results.

Bm No,

Al A2 A3 Bl B2 B3 Cl C2 C3 Dl D2 D3 El E2 E3 Fl F2 F3

Initial Camber due to Prest.

a, bTABLE 4

COMPUTED ULTIMATE MIDSPAN CAMBER, BY TERMS, FOR THE LABORATORY BEAMS AND BRIDGE GIRDERS, USING THE GENERAL

EQS (15), (16), (18) & (20) WITH EXPERIMENTAL PARAMETERS

Initial ccreep carob. ccreep carob. DL Crp Bm DL El def Crp def Defl. up to s 1. cast after s 1. cast defl. up defl. due to due to due to or shk. warp or shk. warp to slab after slab slab Bm,DL up to sl. cast up to sl. cast cast sl, cast DL DL

Laboratorv Beams 0.30 -o. 05 0.37 - -0. 09 - - -0.24 -0. 05 o. 31 - -0.09 - - -o. 19 -o. 05 0.25 - -0.09 - - -0.27 -o. 05 0.34 - -o. 10 - - -0.27 -0. 05 o. 14 o. 07 -0.04 -0.02 -0.05 -0.02 0.27 -0. 05 o. 19 0.05 -0. 05 -0. 01 -0.04 -0.02 0.32 -0. 05 0.40 - -0.09 - -0.32 -0. 05 o. 16 0,08 -0.03 -0. 02 -0.04 -0.02 o. 32 -o. 05 o. 22 o. 06 -o. 05 -o. 01 -0.04 -o. 02 o. 61 -o. 07 o. 69 - -o. 13 - - -o. 51 -0. 07 0.63 - -o. 13 - - -0.47 -o. 07 o. 5 9 - -o. 13 - - -0.49 -0.06 0.58 - -0. 11 - - ~

0.49 -0. 06 o. 24 o. 07 -0.04 -o. 01 -o. 09 -0.03 0,49 -0,06 0.34 o. 05 -o. 06 -o. 01 -0.09 -0.02

- -o. 07 -0.22 - -0. 09 - - -- -0. 07 -0.02 -0.04 -o. 02 -o. 01 -0. 10 -0.02 - -o. 07 -o. 11 -o. 02 -o. 05 -o. 01 -o. 10 -0. 10

De fl due to diff. shk.

----

-0. 0 1 -0.04

--0. 0 1 -0. O·

----

-0,02 -o. 05

--0. 02 -o. 05

Total Camber using Eqs. ( 15 ), ( 16 ), ( 18 ), (2 0)

0.53 0.41 0,30 0.46 0.29 0,30 0.58 0.39 o. 39 1. 10 0.94 0.86 0.90 0.55 0.59

-0.38 -0.30 -0.43

cr-0

a, bTABLE 4 (Cont'd)

Initial Initial ccreep camb. ccreep camb. DL Crp Bm DL El def Crp def De fl Total

Bm Camber Defl. up to s 1. cast after sl. cast defl. up defl. due to due to due to Camber No. due to due to or shk. warp or shk. warp to slab after slab slab diff. using Eqs

Prest. Bm.DL up to sl. cast up to sl. cast cast sl. cast DL DL shk. (15), (16) (18), (20)

Bridge Girders 152 3. 7 1 -1. 56 2. 32 0.68 -1.42 -0. 36 -1.96 -0.79 -o. 17 0.45 15 3 3.87 -1.64 2.39 o. 7 1 -1. 49 -o. 38 -2. 21 -0.89 - 0. 19 o. 17 154 3.87 -1.64 2. 39 0.71 -1. 49 -0. 38 -2. 2 1 -0.89 -0. 19 o. 17 155 3. 72 -1. 5 7 2.29 0.70 - 1. 40 -0. 37 -2.26 -0.91 -o. 19 o. 0 1 156 3. 96 -1.68 2. 38 0.73 -1. 50 -0.40 -2. 01 -0.80 -o. 18 0.50

a All values in the table are in inches.

b See Footnote b, Table 3, for a description of the experimental parameters.

cThe shrinkage warping term and the total deflection term refers to beams with non-prestressed reinforcement only.

62

the girders, after prestressing), and at 180 days for the laboratory

beams and 560 days for the bridge girders. The test period for the

laboratory beams {except Group A) was terminated after 6 months in

order to conduct load-deflection tests. The test period for Group A

specimens was 1 year.

The computed ultimate values are also tabulated in Tables

1 - 2 using the general Eqs. (14) - (18) with experimental parameters

determined for the sand-lightweight concrete of this project, and

using the ultimate-value Eqs. (22) - (27) with general parameters

given for normal weight, sand-lightweight, and all-lightweight con-

crete. For the general parameters, the same creep and shrinkage

factors are suggested for all three concretes, with different modular

ratios and prestress loss ratios {ti F /F and ti F /F ) for each. s 0 u 0

The computed ultimate values for loss of prestress and camber are

shown term by term in Ta bl es 3 and 4 using the general Eqs. { 14) -

(20) with experimental parameters.

4. 6 Discussion of Experimental Results and Conclusion

The experimental and computed loss of prestress and camber

for the lightweight concrete structures of this project are shown in

Figures 10 - 18 and Tables 1 - 4. Results by both general Eqs. (14) -

{20) {for values at any time, including ultimate) with experimental

parameters, and ultimate-value Eqs. (22) - (27) and {30) - (34) with

63

general parameters (given herein) are included. These results serve

to substantiate the generalized procedure presented for predicting

loss of prestress and camber of non-composite and composite pre-

stressed structures. The approximate Eqs. (30) - (34) may be suit-

able for rough calculations only in some cases.

Results computed by the material parameter Eqs. (2), (4),

(7) - (9) are compared with the data of this project in Figures 2 - 7.

Eqs. (2) - (6), (7) - (8) are generalized for different weight concretes.

The procedure for predicting creep and shrinkage is one of providing

standard functions, with suggested ultimate values for different

weight concretes, and correction factors for pertinent conditions

other than "standard" (~). These conditions are briefly described

in the text and Appendix B. The ultimate values suggested should be

used only in the absence of specific information pertaining to local

aggregates and conditions.

Continuous time functions are provided for all needed material

parameters (and for different weight concretes, moist and steam

cured), so that the prestress loss and camber equations readily lend

themselves to computer solutions. Certain other read-in data (such

as for the effect of behavior before and after slab casting-- CLs' 13s•

m, y8

, y , and t>.F /F ) is also included, along with a summary of 81 s 0

parameters convenient for hand calculations. Using these parameters,

64

the calculations needed in the approximate Eqs. (30) - (34) are not

significantly fewer than in the more reliable Eqs. (14) - (20).

It is noted that Eqs. ( 14) - (2 7) could be greatly shortened by

combining terms, but are presented in the form of separate terms

(see results in Tables 3 and 4 and the Sample Calculations) in order

to show the separate effects or contributions to the behavior (such

as due to the prestress force, dead load, creep, shrinkage, etc.,

that occur both before and after slab casting).

The following specific observations and conclusions are made

J'elative to the results in Figures 8, 10 - 18, Tables 1 - 4 and other

parts of the report:

1. The ultimate steel relaxation percentage recommended

for regular 7-wire strand to be used in prestressed concrete struc

tures is 7. 5%. See the results and discussion of Figure 8, Term (4)

of Eq. (14), and References (45) and (46).

2. The computed initial camber agreed well in most

cases with the measured initial camber, as shown in Table 2.

3. The computed prestres s loss for the laboratory non-

compos ite beams was varied (from -1.4% to 2.8% prestress loss

differential after 6 months) from the experimental results (see

Figures 10 - 12 and Table 1). The direct application of laboratory

creep data for uniformly loaded specimens to beams with non-uniform

65

stress distribution appears to slightly overestimate the creep effect.

The same effect, however, was not noticed in the camber results.

This is probably due to the fact that in the loss computations, the

F/A stress component is a dominant factor while in the camber

computations, there is no corresponding deforn1ational comp:ment.

Other prestress loss and camber results in Figures 13 - 18, and

Tables 1 and 2 are considered to be in very good agreement. For

these cases (non-composite beam camber and composite beam loss

and camber), offsetting creep (and shrinkage in the case of com

posite beams) effects occur.

4. As shown in Figures 10 - 12 and Table 1 the difference

in the end and midspan prestress loss was quite small for the labora

tory beams, and relatively large for the bridge girders before slab

casting. After slab casting, the prestress loss in the bridge girders

was only slightly different at end and midspan.

5. The loss of prestress for the sand-lightweight con-

crete bridge girders was of the order of 27% to 29% at 560 days after

prestressing and 29% to 31% ultimately (see Figure 13 and Table 1).

It seems clear that loss percentages for bridges under similar condi

tions using normal weight concrete will normally be somewhat lower

than these (of the order of 25%); and using all-lightweight concrete

will normally be somewhat higher than these (of the order of 35%

or higher).

66

6. Slab casting causes an elastic deflection (downward)

and prestress gain, and a time-dependent deflection and prestress

gain, due to creep and differential shrinkage. Loss of prestress due

to creep and camber growth under the prestress force and precast

beam dead load are also reduced by the effect of the hardened slab

(as opposed to the case of no composite slab). These results can be

seen in Tables 3 and 4 and the Sample Calculations. The composite

slab reduces the ultimate loss of prestress at midspan of the bridge

girders about 11% (as 41% - 30% = 11%). It can be seen in Figure 18

and Table 4 that the camber curves have nearly levelled off at about

3. O" just before slab casting. After slab casting and up to ultimate,

the camber is reduced to near zero.

7. The effect of the 3-week and 9-week slab casting

schedules for the laboratory beams had only a small effect on loss

of prestress (Figures 11 and 12) and a more noticeable effect on

camber (Figures 15 and 16). When considering a 3-week slab (slab

cast 3 weeks after prestressing) for the bridge girders, as compared

to the actual 9-week slab, the ultimate loss of prestress at midspan

was about 2% less and the ultimate midspan camber about O. 10" less

for the 3 week case. These results serve to point out the relatively

small beneficial effect of casting the deck slab as early as possible

(also indicated in Reference (li)). It is noted that there are also

67

offsetting effects in the case of the effect of slab casting schedules.

An earlier slab tends to reduce total creep deformation (causing

upward camber) by forming an earlier composite section, but also

reduces differential shrinkage deformation (causing downward

deflection).

8. The different individual contributions to prestress

loss and camber, as illustrated by the different terms in Eqs. ( 14) -

(29), are sensitive to the stiffness, creep, and shrinkage concrete

properties. However, the net results of these equations tend toward

more correct solutions than the individual terms because of off

setting effects. This is especially true in the case of composite

beams, and is less the case for non-composite beams, See Tables

1 and 2 and the comparison of ultimate-value results with experimen

tal parameters and general parameters.

9. The inclusion of all terms inEqs. (14) - (29) appears

to incorporate all significant effects in the reliable prediction of

prestress loss and camber. These effects can be seen in the term

by-term tabulations in Tables 3 and 4, and the Sample Calculations.

In the sample calculations for the bridge girders using the general

parameters, for example, the 7 terms (omitting differential shrink

age--Term 8) for loss of prestress varied from 1. 6% to 12. 7%, and

the 9 terms for camber varied from 0.48" to 4.09". The results by

68

the approximate Eqs. (30) - (34) and the more reliable equations

were in reasonably good agreement (see Tables l and 2 and the

Sample Calculations) for most of the structures of this project.

10. All of the bridge girder data in Figure 18 showed an

increase in camber of about 0. 4" between 300 to 370 days (starting

in April). This appears to be due to higher temperatures and is

consistent with the observations of Delarue (~i)•

11. The systematic procedures described in this paper

for predicting time-dependent behavior are deterministic in nature.

Probabilistic methods are also needed for estimating variability

of behavior.

12. Sand-lightweight concretes using Haydite (as the

coarse aggregate) show slightly higher creep (Cu = 2. 00) than sand-

lightweight concretes using Idealite (as the coarse aggregate)

(C = 1. 75) under identical loading and environmental conditions, u

(Figures 3 and 6).

13. There does not seem to be any fundamental difference

between all-lightweight Haydite concrete and sand-lightweight Hay-

<lite concrete as far as the creep properties are concerned. (Figure

6). The loss of prestress for beams made of all-lightweight Haydite

concrete is substantially greater than for beams made of sand-light-

weight Haydite concrete (Figure 12 and Tables 1 and 3). This is

69

due to the high elastic deformation of all-lightweight concrete (due

to its low elasticity modulus) and not due to the difference in creep

behavior of the two concretes.

14. The effect of the 4-week and 10-week slab casting

schedule for the laboratory reinforced beams had a very noticeable

effect on deflection (Figure 17). An earlier slab tends to reduce

total creep deformation (causing downward deflection) by forming an

earlier composite section, and also reduces differential shrinkage

deformation (also downward deflection). When considering a 4-week

slab (slab cast at beam age = 4 weeks) for the laboratory beams

(reinforced), as compared to the 10-week slab (slab cast at beam

age= 10 weeks), the ultimate deflection was about 0. 13" less for

the 4-week case. These results serve to point out the relatively

large beneficial effect of casting the deck slab as early as possible

for reinforced beams.

15. In comparing non-composite reinforced beams with

composite reinforced beams, it is noticed that the ultimate deflection

of the non-composite beam was about 0. 08" greater than the 4-week

case, but about 0. 05" lesser than the 10-week case. The earlier

composite section (4-week slab) reduces the total deformation by its

composite action, while the later composite section (10-week slab)

increases the total deformation due to the various shrinkage effects

70

(Table 4). This effect, however, may become very small in regions

of high humidity.

4. 7 Comparison of Computed and Measured Data Reported by

Others (Q), (24), (~2), (I.!:_)

Simultaneously measured deflections and strains of prestressed

concrete beams reported in the literature are scarce. The strains

and deflections reported by Branson (Q) were taken from post

tensioned beams. Both composite and non-composite beams were

included in the study. Unit creep curves of the concrete were not

reported. The total strains of the beams were measured and

reported. A reciprocal approach can be used from these strains to

arrive at a value of the ultimate creep and shrinkage coefficients.

The report of Corley, Sozen, and Siess (24) and Sinno (27) included

all the relevant information required to perform the predictions by

methods presented in this paper. Pauw and Breen (l.!:_)have reported

the strains and camber measurements of two post-tensioned com

posite bridge girders. Separate creep tests are not included in this

report. The experimental loss of prestress is determined from the

measured concrete strains in a manner similar to that described in

Figure 9. The loss due to steel relaxation is as given by Term (4)

of Eq. (14).

Prediction of loss of prestress and camber for the beams in

71

References (~~), (24), (~) and (2-!_) are obtained using the general

Eqs. (14) - (20) and experimental parameters (where available) and

the general parameters mentioned in this report, and each is com

pared with the measured results.

Results of Tests at the University of Florida (~)

Description of Specimens :

Ten post-tensioned normal weight concrete beams of spans

19'-6" were cast and studied for a period of about 5 months for both

camber and loss of prestres s. Eight of these were stored in the

laboratory and the other two were stored in the field. Cast-in-place

slabs were cast on five of the beams at ages varying from 37 to 101

days. The properties of the test specimens are shown in Table Cl.

Shrinkage specimens were also cast.

Discussion of measured and computed results:

The results for the loss of prestress (at end and midspan)

as well as for the midspan camber are shown in Figures 19 - 21,

using both the general parameters (suggested in this report) and

the experimental parameters (estimated from reported strains).

From this comparison (Figures 19 - 21), the following observations

are made:

1. The general parameters being slightly smaller than

the experimental parameters tends to underestimate the loss of

(A) Using exptl. parameters 30.---~~~~~-'-~-r---~--.-~~---,,-~~

.&Bm 1 • Bm 6 • Bm 2 <J Bm 7 O Bm 3 £Bm 8

I::;. Bm 4 ii Bm 9 DBm 5 Lli.Bm 10

201--=.::.::.;--=-.=.::;:..:.:.:__::-=+-f-'-,,~o-,"4----,~

END

0 10 20 30

(B) Using general parameters 30~'--'--~~"'--'"--~~~°"-~~~~~~~

ABm 1 • Bm 6 e Bm 2 <J Bm 7 0 Bm 3 £Bm 8 l:;.Bm 4 ii Bm 9 DBm 5 4Bm 10

20!-=:....::'.~.::_=..c:;:..:.:=--..::_:::++_.-~-1--.,.-,.,...,.1<"-----I

END

0 10 20 30

Experimental loss of prestress in percent of initial stress

Figure 19 Computed and experimental loss of prestress at end of beams reported in Reference (~)

.... N

30 (A) Using exptl. parameters

0

ABm 1 • Bm 6 • Bm 2 ct Bm 7 O Bm 3 £Bm 8 l:> Bm 4 l.i Bm 9 o Bm 5 A Bm 10

10

CENTER

20 30

30 (B) Using general parameters

ABm 1 •Bm 6 • Bm 2 ct Bm 7 O Bm 3 -'ilBm 8 !:; Bm 4 i.i Bm 9 DBm 5 dBm 10

10

CENTER

20

Experimental loss of pres tress in percent of initial stress

Figure 20 Computed and experimental loss of prestress at center of beams reported in

Reference (2 3)

30

'"" "' ::; P-< s 0 u

(A) Using exptl. parameters 1,2 r'--'--~~--"'---'"'-'-C..::..:--"'c-"-''-"':C::.::...::..::..::..::.._~~~ A Bm 1 • Bm 6 • Bm 2 <t Bm 7 O Bm 3 ~Bm 8 6 Bm 4 Iii Bm 9 o Bm 5 dBm 10

0,31----=-=;c_::_-=-_:;::..:_::__::_:_1--,,._-l-__Jm..<:.i.-~---1

0 0.4 0,8 1. 2

(B) Using general parameters l.2r-~~~~"--~~r-=--~~~~~~~

A Bm 1 • Bm 6 • Bm 2 <t Bm 7 0 Bm 3 .&Bm 8 £:,. Bm 4 Iii Bm 9 o Bm 5 4Bm 10

0.81--~---.------'=-r--~A-~~+--~-+-~__.L1

OIL.~--1..~~.L-~-1...~~.L-~-'-~--' 0 0,4 0,8 1. 2

Experimental values of midspan camber in inches

Figure 21 Computed and experimental midspan camber of beams reported in Reference (fl)

75

prestress and camber. The scatter between the measured and com

puted loss of prestress, using experimental parameters, is + 15%,

while the same, using general parameters is.± 20%.

2. The scatter between the measured and computed values

of camber, using experimental parameters, is.± 15%, while the same,

using general parameters is +30%. The increase in scatter of+ 15%

for camber and only+ 5% for loss of prestress (using general param

eters) suggests that camber is more sensitive to changes in param

eters than loss of prestress.

3. The computed initial values of camber agrees very

well with the measured values for all of the beams.

Results of Tests at the University of Illinois (24)

Description of Specimens:

Two pretensioned non-composite rectangular beams of

normal weight concrete and 6' spans were observed over a period

of two years under laboratory conditions. Midspan camber and

strains were recorded periodically. The properties of the test

beams are shown in Table CZ.

This paper includes all the relevant information pertaining

to elastic properties, creep, and shrinkage characteristics, that

are needed to perform the predictions presented in this study for

the loss of prestress and camber.

76

Discussion of Measured and Computed Results:

The results for the two beams for loss of prestress (at center

only) and midspan camber are shown in Figures 22 - 23. From these

comparisons, the following observations are made:

1. The general parameters (suggested in this report)

being smaller than the experimental parameters causes an under -

estimation of the loss of pre stress. Part of this underestimation is

due to the variation in the value of the modulus of elasticity at

release (due to the use of Eq. 6) from the measured value. The

scatter between the measured and computed loss of prestress along

with the computed values of Eci (using Eq. 6) is + 10% for the experi-

mental parameters and_:!: 15% for the general parameters. However,

the use of the measured values of Eci reduces the values of scatter

for the general parameter results by + 5% (Figure 22). This indicates

that differences between the measured and computed values of the

modulus of elasticity at release should not be overlooked.

2. The effect of the smaller general parameters is

significantly felt on the values of camber. The scatter between the

measured and computed values of camber along with the computed

values of E . (using Eq. 6) is + 20% for the experimental parameters Cl -

and_:!: 35% for the general parameters. However, the use of the

measured values of Eci reduces the values of scatter for the general

parameter results by_:!: 10% (Figure 23).

tll tll tll <!) tll

'" <!) ..... '" tll ..... <!) tll

'" -p. "' '-+-! ·n 0 .....

~ tll .....

tll 4-<

.3 0

"d <!)

";j p.

8 0 u

A) Usin arameters 60F"'~~=>--====-'-':;=-::.:=..:.:.:..;.:=-=c=---,,,_,.,

.6.MUl DMU2 4M \.., 1] us ing Ill M U2 meas • E . t----tti.-7'-r-t-,..._,,,._--1

Cl Range usin meas. Eci

401--~-r""'-'==-T-----""'---l--~.<+~'---+-~---I

CENTER

0 20 40 60

60 (B) Us in eneral parameters

.6.MUl DMU2

'1.M U l] using Ill MU2 meas. Eci

Range usin meas. Eci

401--~~~~~--'"-"--f--,~-

CENTER

0 20 40

Experimental loss of pres tress in percent of initial stress

Figure 22 Computed and experimental loss of prestress at center of beams reported in Reference (24)

60

(A) Using exptl parameters 0.24.r-~~~~~~~~~~~~~~~

., AMUl o MU2 ~-,

~ 4MU1] using ' .S Iii M U2 meas. E ci t----,l'-+--71l-.,.._-

>1 Range ·~

0.08 o. 16 0.24

(B) Using general parameters 0.24~~~~~~~~.c-~~~~~~~

AMUl CJ MU2

4MU1] using Iii MU2 meas. Eci

Range

0 16 1----;;m~e~a~s~·,.:::~L..,1'-----lf----,!,..,...4il~ .

0 0.08 o. 16 0.24

Experimental midspan camber in inches

Figure 23 Computed and experimental midspan camber of beams reported in Reference (24)

__, 00

79

3. The computed initial values of camber agree fairly

well with the measured initial values of camber, though the magni-

tudes are very small. It is, however, noted that (due to the initial

values of camber being very near zero) a small deviation from the

measured value at the initial stage is reflected in a larger magnitude

at a later stage. Even though the scatter between the measured and

computed values of camber using general parameters along with the

computed value of E . is + 35%, the actual difference between the Cl -

computed and measured camber is less than 0.06" (Figure 20).

Results of Tests at the Texas A & M University (~)

Description of Specimens:

Five non-composite pretensioned Type B bridge girders of

the Texas Highway Department (4 lightweight and 1 normal weight)

of spans 38'-45 1 were studied over a period of 1 year for both

camber and loss of prestress. The girders were maintained in the

field. The properties of the specimens used in this study are shown

in Table C3. Standard 6 11 by 12" cylinders were cast and used to

determine the strength of concrete. In addition to the five girders,

2 shrinkage specimens (of the same cross section as the girder)

but 4' long were also cast.

This pa.per includes all the relevant information pertaining

to elastic properties, creep, and shrinkage characteristics, that

80

are needed to perform the predictions presented in this study for

the loss of prestress and camber. While using the general param

eters, the correction factors were extrapolated for conditions other

than the "standard" (see Chapter 2).


The results for the loss of prestress (at end and midspan)

and the midspan camber are shown in Figures 24 - 26 for the five

girders. From these comparisons, the following observations are

made:

1. The general parameters (suggested in this report)

being slightly greater than the experimental parameters overesti

mates slightly the loss of prestress and camber. The scatter between

the computed and measured loss of prestress at the end of the beam

using the experimental parameters and the general parameters are

+ 16% and.:!: 20% respectively. The corresponding values at the

center of the beam are .:!: 15% and.± 20% respectively. The difference

between the experimental parameters and the general parameters is

noticed in the slight increase of scatter for the latter case. This

increase is, however, small and within the tolerances of design.

2. The computed values of initial camber agrees fairly

well with the measured values.

3. The scatter between the measured and computed values

..... ~ 30 (A} Us in exptl. parameters (B} Using general parameters

30,---~~~~~~~-r-~~.--~,,,r-~--,

u

'" ~ AL1-5•L3-5

.8 e L4-5 6Rl-5

•

END

o~~-'-~~"-~-'-~~~~~~___,

0 10 20 30 0

A. Ll-5 • L3-5

e L4-5 6Rl-5

o R4-5

10


END

20

Figure 24 Computed and experimental loss of pres tress at end of beams reported in Reference (~2.)

30

1:l ., u ... ., p.

parameters

AL1-5•L3-5 Oe 30

(B) Using general parameters

ALl-5 •L3-5

.: ..... e L4- 5 .6. R 1-5 f--i!!'P--,e~---1 e L4-5 .6.Rl-5

O R4-5 "' "' ., ~ 2 Ot---.----,,-----l""4~3-t-,1i...,""-l---I .t: ., "' ... t ~ p. ..... ~ -~ 0 .... ..... .. .: "' ..... 101----1--~~f-.~--le----l----l----I 0 ....

..... 0 "Cl ., .... ::l p.

8 0 u

0

CENTER

10 20 30

o R4-5

CENTER

0 10 20


Figure 25 Computed and experimental loss of prestress at center of beams reported in

Reference (~1)

30

OJ N

"' 2 • 4 (A) Using exptl. parameters

" -5 A Ll-5 • L3-5 q

·n 2, 0 e L4-5 .6. Rl-5 q ·n

H 0 R4-5 JS 1 . 6 t----,---,--j--,.£..---f''----7"-~t----i s ro u q l.2t---+---t--.AJ..t----,?.-t---+---I ro P;

"' "Cl ·g o. 8

"Cl

" ~ 0,41----LllP".---+---+---+---J----j P;

s 0 u

0 o. 4 o. 8 1,2 1.6 2.0 2.4

(B) Using general parameters 2.4.-----~~--.=----,---,,-----,

A Ll-5 • L3-5

2. 0 e L4-5 .6. Rl-5

O R4-5

0 0,4 0,8 1. 2

•

1.6 2,0 2.4

Experimental camber at midspan in inches

Figure 26 Computed and experimental midspan camber of beams reported in Reference (~Z.)

00 w

84

of midspan camber using the experimental parameters and general

parameters are..± 15% and+ 20% respectively. The sensitivity of

camber computations to the choice of general parameters is noted.

An increase of + 5% in scatter for the use of general parameters

is considered reasonable.

Results of Tests at the University of Missouri (~)

Description of Test Specimens:

Two post-tensioned prestressed composite beams of normal

weight concrete and spans 99' were observed over a period of two

years under field conditions for camber and loss of prestress.

Concrete strains were measured at both end and midspan for

both the beams. The properties of the test girders are shown in

Table C4.

This paper does not include any information pertaining to

the creep and shrinkage characteristics of the concrete. An arbi-

trary value of C = 3. 00 was used in this paper for the computation u

of camber. To obtain an idea of the range of behavior of this girder,

the measured values of the loss of prestres s (at end and midspan)

are compared with the computed values of loss of prestress (using

maximum and average general parameters). A similar comparison

is made between the computed and the measured values of midspan

camber (using maximum and average general parameters).

85

Slabs were cast on these girders at pre cast beam age = 200

days. However, the value of ~s as computed by Eq. 10 is based on

data of specimens whose loading ages are not more than 50 days.

As there is no available literature for later loading ages (200 days),

an estimated value of ~s is used in all the computations for the loss

of prestres s and camber.


The results for the loss of prestress and camber are shown

in Figures 27 - 29 for the east girder. The results for the west

girder cannot be computed with the limited information available in

the paper. In the computation of initial values of camber, numerical

methods were used (to account for the variable moment of inertia).

From these comparisons, the following observations are made:

1. The use of maximum general parameters overesti-

mates the loss of prestress at both end and center by 20% and 25%

respectively.

2. The use of average genel"al parameters estimates

reasonably well the loss of prestress at both end and midspan (scattel'

of + 10% for both).

3. The use of maximum general parameters estimates

the midspan camber very well (scatter of_! 10%).

4. The use of average general parameters results in a

"CJ

"' .., ::J p.,

8

30(A) Usin

A East girder

30 (B) Using avg. eneral parameters

A East girder

END END

0 u OIL..~-L~~.L.._~-1...~~~~~~__J

0 10 20 30 0 10 20

Experimental loss of pres tress .in percent of initial stress

Figure 27 Computed and experimental loss of prestress at end of beam reported in

Reference (~.!)

30

(A) Usin max. general parameters 30

.A. East girder ~· ,. ...

CENTER

O"'-~--'~~-'-~~-'-~~~~~~~~

0 10 20 30

(B) Using avg. general parameters 30

.A. East girder

CENTER

0 10 20


Figure 28 Computed and experimental loss of prestress at center of beam reported in Reference (2.!_)

,

30

'O Q) +> ::J P< s

q ....

(A) Using max. general parameters o. 6

.... East girder

0.21--~~1--~-.Yf-"~--t~~--t~~--t~~---<

8 o"-~---'~~--'-~~-'-~~.L.....~......J~~_J 0 0.2 0.4 0.6

(B) Using avg. general parameters o. 6

.... East girder .-, ~· I

Experimental values of mids pan camber in inches

Figure 29 Computed and experimental midspan camber of beam reported in Reference (l!_)

00 00

89

difference between the computed and measured values of midspan

camber of 30%. It should, however, be noted that in spite of the

wide difference of 30%, the actual difference for the worst data point

is less than O. 18". Realizing that this difference between the com

puted and measured values of camber is for a girder of about 100'

span, the difference of 30% has only an academic significance.

4. 8 Summary of Results Reported by Others and Conclusion

On the basis of Figures 19 - 29, and the specific conclusions

made in section 4. 7, the following general observations are made

concerning the design method suggested in this report and the experi

mental results of University of Florida (~), University of Illinois

(24), Texas A & M University (27), and University of Missouri(~_!):

1. The use of the average general parameters and the

general Eqs. ( 14) and ( 17) is a reasonable means of computing the

loss of prestress for both composite and non-composite beams.

Either an underestimation (Figures 19, 20, 22) or an overestimation

(Figures 24, 25, 27, 28) may occur, depending on the difference

between the experimental and general values of the creep and

shrinkage parameters. However, the maximum scatter between

the computed and the measured values of loss of prestress was

+ 20% (using average general parameters) for these studies.

90

2. The use of the average general parameters and the

general Eqs. (15) and (18) is a reasonable means of estimating mid

span camber for both composite and non-composite beams. Either

an underestimation (Figures 21, 23, 2 9) or an overestimation (Figure

26) may occur, depending on the difference between the experimental

and general values of the creep and shrinkage parameters. The

maximum scatter, however, between the computed and measured

values of midspan camber is_± 30% (using average general parameters).

This maximum value of scatter occurs only in 3 of the 18 beams

studied (Figures 23, 29) and even in these cases, the difference

between the computed and the measured value of camber is less than

O. 18 ". The scatter between the computed and measured values of

midspan camber for the remaining 15 beams is _± 25%.

3. The procedure suggested in this report for the predic-

tion of initial camber is adequate.

4. Camber computations are more sensitive to the choice

of creep and shrinkage parameters than loss computations for non

composite beams. The reverse is true for the composite beams

because of the offsetting effects that may result in "near zero" camber

or deflection values after slab casting. These offsetting effects are

primarily due to the elastic and creep deflections due to the slab

dead load, and increased stiffness of the section on the one hand as

91

opposed to the reduced prestress force and its creep deformation on

the other.

5. The choice of the value of the initial modulus of

elasticity can affect the loss of prestress and camber (see results

of tests at the University of Illinois). In fact, the value of E . Cl

affects camber more than the loss of prestress.

6. It is reasonable to expect that the use of general

parameters along with the approximate Eq. (31) (for ultimate loss

of prestress) and Eqs. {30) and (32) (for ultimate midspan camber)

will result in values slightly higher than those obtained by the use

of the ultimate Eqs. (22) to (27).

92

Chapter 5

LOAD-DEFLECTION STUDIES OF PRESTRESSED AND REINFORCED CONCRETE BEAMS

5. 1 General

Increasing interest is being shown in the design of prestressed

concrete members that crack under working loads. Since substantial

cracking occurs under working loads in ordinary reinforced concrete

members, cracking at service load levels in prestressed concrete

members should be acceptable provided appropriate safety and ser-

viceability requirements are met.

This chapter is devoted to the study of prestressed concrete

beam deflections under a single load cycle (a single cycle is defined

herein as a continuously applied increasing load to failure at a static

rate) and repeated load cycles, and reinforced concrete beam deflec-

tions under increasing loads and 24-hour sustained cracking loads.

Both rectangular and composite T-beams are included.

The details of the test beams are shown in Tables Al and A2.

The concrete properties of the laboratory beams at the time of the

load-deflection tests are shown in Table A6. The laboratory beams

were tested as follows:

93

Groups A, B, and D: Single -cycle load tests for pres tressed beams

Group C:

Group E:

Group F:

Repeated load tests with constant load cycle for pres tressed beams

Repeated load tests with increasing load cycle for prestressed beams

Increasing load and 24-hour sustained load tests for reinforced beams

Observed midspan deflections shown in Figures 32-34, 37-48

refer to the position of the beam just before the application of the

transverse load. If the deflections from the positions of the beams

before prestressing are desired, the initial camber under prestress

and dead load and the time-dependent camber must be subtracted

from the deflections in Figures 32-34, 37-48. A two-point loading

system {Figure 30) symmetrical about the centerline of the beam was

used in all of the tests.

5. 2 Single Cycle Load Tests of Pres tressed Members

Deflection of uncracked members

The elastic theory can be accurately applied to concrete beams

as long as the concrete is not cracked. Distinct changes occur in the

behavior of concrete members after first cracking. After cracking,

there is a change in the distribution of bond and shearing stresses

and the load-deflection response changes sharply.

94

The determination of cracking loads can be based on the elastic

theory, assuming that cracking starts when the tensile stress in the

concrete reaches its modulus of rupture. The accuracy of the elastic

P/2 P/2

5 1 -6 11 5 1 -6 11

15'-0"

Figure 30. Two point loading for 'load-deflection' studies of laboratory beams

theory and also the modulus of rupture obtained from the usual bending

tests as being representative of the tensile strength of concrete in

bending has been questioned (48). However, most available test data

indicates that the use of the elastic theory up to cracking (determined

with the modulus of rupture) is sufficiently accurate.

For a pres tressed concrete beam without non-tensioned steel,

the cracking moment is given by:

Ftlg

AgYt

where Ft = Fi - l\Ft; Fi is the initial prestressing force and

l\Ft is total loss in prestressing force obtained by

using Eq. (14) or (17).

(35)

95

Ag = gross area of section

I = gross moment of inertia of section g

Yt = distance of tension fiber from cgc

' fcb = modulus of rupture of concrete.

Shaikh and Branson (49) indicated that the cracking moment

of prestressed concrete beams is (for all practical purposes) not in-

fluenced by the addition of non-tensioned steel. It was concluded that

Eq. (35) may be used to compute the cracking moment of prestressed

concrete beams containing non-tensioned steel in addition to pre-

s tressing steel.

Deflection of cracked members

Under cracked conditions, the behavior of prestressed concrete

members and ordinary reinforced concrete members is similar.

Since ordinary reinforced concrete members are invariably cracked

under working loads, most methods for computing these deflections

do _take into account the effect of flexural cracking in some form.

For this investigation, the method of Brans on ( _! )(2.2_)(2.!_)(42)

was used to compute the deflections of the test beams. The choice

of this method (Eqs. (37) and (38)) is based on favorable comments

from designers and on its indicated accuracy in the A Cl Committee

435 report(_!) on deflections of reinforced concrete flexural mem-

bers. These have been proposed for the 1971 ACI Code (50}(2.!_).

96

For an elastic homogeneous member subject to flexure:

M CD= EI (36)

The curvature, cp, at any section can be readily obtained using Eq.

(36), with the appropriate bending moment, M, and flexural rigidity,

EI, at that section. For uncracked sections either the gross, or,

more precisely, the uncracked transformed moment of inertia may

be us ed. Under cracked conditions, however, because of the varying

amount and extent of cracking, the flexural rigidity, EI, is not a

constant.

Theoretically one could evaluate zones for which the cracking

moment is exceeded and thus calculate the corresponding transformed

section moments of inertia along the length of the beams, based on

appropriate cracked and uncracked sections. With the flexural

rigidity known along the length of the beam, curvatures could be com-

puted using Eq. (36) and deflections obtained by the usual procedures.

Due to the complexity involved in relating the height of cracks,

spacing of cracks, etc. to the flexural rigidity of the member, mostly

empirical or grossly approximate methods have appeared in the liter-

ature for computing flexural rigidity, EI, under cracked conditions.

Based on a sizable number of tests on rectangular beams

(simple and continuous) and T-beams, Branson (50) has presented

an empirical expression for the effective moment of inertia at a

97

given section, Ieff• The expression was given in a form that includes

the effect of extent of cracking as:

(3 7)

where: Mer = cracking moment as defined by Eq. (35)

M = bending moment at the section where Ieff is desired

I = moment of inertia of gross section g

Icr = moment of inertia of the fully cracked section using

Eq. (39). See Figure 31.

An expression for an average effective moment of inertia for

the entire length of the simply supported beam under uniformly dis -

tributed load was also given by Brans on (~) as:

+ (38)

where: Mmax = maximum moment in the span.

It is to be noted that Eqs. (37) and (38) apply only when Mor

Mmax is greater than or equal to Mer; otherwise Ieff =lg. For con-

tinuous beams, the average of positive and negative moment region

values in Eq. (38) is recommended (42)(50)~).

98

b b(kd)

3 ·I , ..

I = +nA (d-kd)2

kdl __

er 3 s

(np)2 k = + 2np - pn d A

n = E /E s where: --- s c

p =A /bd s

Figure 31 Moment of inertia of cracked section (I ) er

-·-.,,

_, .,,

{39)

The concurrence of AASHO, ACI, and PCI codes on the methods

of determination of ultimate strength of prestressed concrete beams

establishes the reliability of the equations indicated in the codes.

Therefore, in this investigation only a comparison of observed and

computed (using equations from the ACI code) values of ultimate load

was obtained.

Single cycle load tests were conducted on all the beams of Grps.

A, B, and D. Midspan deflection of the test beams were obtained up to

loads ranging from 76 to 88 percent of the ultimate loads. Eq. (38)

''The same equations are also valid for composite beams (with trans -formed compression flange width to account for the different concretes) if the neutral axis falls within the flange. This was the case for the laboratory composite test beams studies herein.

99

was used to determine the effective moment of inertia in the computa-

tion of deflections. Eq. (35) was used for computing Mer• and Eq.

(39) was used for the determination of Icr· The modulus of rupture,

I fcb' was obtained by bending tests on plain concrete specimens for

the test beams. It is observed that Eq. (38) was originally established

for use in the case of simply supported beams under uniformly dis-

tributed loads. Its use, however, is considered adequate for the two-

point test loading.

The comparison of observed and computed midspan deflection

curves are shown in Figures 32 to 34. Table 5 shows the computed

and measured values of ultimate loads as well as the maximum dis -

crepancies in the observed and computed deflection curves.

Based on Figures 32 to 34 and Table 5, the following cibserva-

tions are made:

1. There are three distinct stages of behavior in the load-deflec-

tion history of a prestressed concrete beam. In the first stage, the

curve is virtually linear. This stage represents the behavior of the

beam before cracking of the concrete. The extent of this stage

depends on the geometrical and material properties of the section

and the type of loading. In the second stage, the load-deflection

curve is characterized by a constantly changing rate of deflection

with applied load and represents the behavior of the beam after the

9,0 ---- Computed Beam Al

Observed \

_\ -~ -- -£:::, Beam Al

8.0

- . - A D Beam AZ

.& ~ ~ L---Beam A3 -· "'"Beam AZ -\,.--___........ . -

~ ... -v -~;-: ~ Beam A3

--~ . .. - C1

v ~ ~- --.. l:;::ii: .. -

A• , .. -

~ ~ v Crack. load Ulti. load •:'For details of , Bm Type i,~ ~.<

~ Com TI Meas Com1 Meas computation, ,,, v Al Rect 3. s{ 3. 40k 8. off 8. 7rf

refer to text.

7. 0

6.0

5.0

4.0

3.0

z. 0 AZ Re ct 3. 04k 3, orf 6. 56k 6. sd< A3 Re ct z. 61' z. 60k 5. zf 5, 3{-

~

~ 1. 0

0 .

0 o. 3 0,6 0.9 1. z 1.5 1. 8 z. 1 Z.4 z. 7 3. 0 3.3 3.6 3. 9 4. z Midspan deflection in inches

Figure 3Z Observed and computed midspan deflection versus load curves for beams of Group A (three non-composite pres tressed beams}

~

0 0

'" P; •M

'""' " •M

'"Cl ro 0 ~

'"Cl

"' •M ~

P; P;

""

9. 0 '

8. 0

7.0

6. 0

Beam B2 /

........___~ Y"; rj/

Crack. load Ulti. load ':'For details

Beam B3 Bm Type Com;'.' Com;:

of computa-//~ Meas Meas refer P' 6. 40k

ti on, Jl Bl 3. or!' k 6. 43k Re ct 2. 92 to text.

f 5. 7rf 5. 65k k k B2 T Bm 9. 34 9.46

B3 5. 6? 5. 55k k 9. 4.f<

' T Bm 9.34

-5. 0

4. 0

3.0

-. - - l'. Beam Bl - - .A L--

r..::---~ --_..... •• ,,.-v ~

& .L.

----- Computed

2.0 Observed

1. 0 J 6. Beam Bl

0 Beam B2

r/ .. Beam B3

0 0 0.2 0.4 0.6 0.8 1. 0 1.2 1. 4 1. 6 1.8 2.0 2.2 2.4

Midspan deflection in inches

Figure 33 Observed and computed midspan deflection versus load curves for beams of Group B (one non-composite and two composite prestressed beams)

9. 0

8. 0

7.0

-----Computed Beam J:?l

' Observed

\ --f:::, Beam Dl ----- 0 Beam DZ ---- -0 Beam D3 ~

Beam DZ - ---- ...... --- --- ,_ -6. 0

"' P< ·~ .-'4 5.0

>1 ·~

"Cl rd

4.0 0 -"Cl Q) .~ - 3. 0 P< P<

"' z.o

--- "" - j --~ ~ .,,.,.. __ -

~ ~""' ~~ ~ - ~BmD3

-~ ~ ~ -~

..... L v-- . 0, Bm Type Crack. load Ulti. load ,,For details of

.... om~ Com:: computation,

Meas Meas

F refer to text .

k 3.85k 8. 58k 8. Z9k Dl Re ct 3.94

DZ Re ct 3. 4rf 3. Z5k 7. 36k 7.6lk

1. 0

0

) D3 Re ct 3. 1 {- 3.00k 6. 8Zk k o.84

I . . '

.

0 0.3 o. 6 o. 9 1. z 1. 5 1. 8 Z.l Z.4 Z.7 3.0 3.3 3.6 3.9 4.Z


Figure 34 Observed and computed mids pan deflection versus load curves for beams of Group D (three non-composite pres tressed beams)

-0 N

TABLE 5

WORKING LOAD, COMPUTED AND OBSERVED VALUES OF ULTIMATE LOAD AS WELL AS VALUES OF WORST DISCREPANCY BETWEEN

COMPUTED AND OBSERVED DEFLECTION CURVES

Group No. A B D

Beam No. Al A2 A3 Bl B2 B3 Dl D2

aComputed Ultimate load, Pu 8. 08 6. 56 5.27 6.43 9.35 9.35 8.58 7. 36 kips .

Measured Ultimate load, pmn 8.70 6.58 5.37 6.40 9.46 9.47 8. 29 7.61 kips

bworking load, PW (kips) 3.57 3. 04 2.67 3. 08 5. 7 1 5.66 3.94 3.40

Load factor,· P uf Pw 2.26 2. 16 1. 97 2. 08 1. 65 1. 65 2. 18 2. 16

cp max (kips) 7. 10 5. 15 4.25 5. 15 8. 18 7.81 6. 5 0 6.oo

(P /P )100 max u 88% 80% 81% 80% 88% 84% 76% 82%

d Pc, (kips) 7.00 5. 00 4.00 4.00 1. 00 1. 00 6.00 3.00

e Worst discrepancy in -12% -12% -21% -24% -14% -13% -5% +8% deflection curves

D3

6. 83

6.84

3. 17

2. 15

5.25

77%

5.00

-10%

aThe computation of ultimate loads is based on accepted procedures indicated in ACI 318-63 Code. The corresponding equations are not reproduced here. The test period varied between 45-60 min for each beam.

TABLE 5 (Cont'd)

bFor the test beams, the working load was assumed to represent the condition that cracking would occur as soon as this load was e:i<ceeded. These values of P w were the computed cracking loads.

c Represents the maximum load for which deflections were recorded.

d Represents the load at which the discrepancy between the observed and computed deflection is the greatest.

ePlus or minus indicate that computed deflection is greater than or smaller than the observed deflections.

105

concrete is cracked and while the reinforcement stress is still in the

'elastic' range of the stress-strain curve for the reinforcement. The

third stage is marked by a very slow change in the slope of the load

deflection curve. In this stage, the reinforcement stress is in the

'inelastic' range of the stress-strain curve for the reinforcement

and the load-deflection curve is nearly flat.

In addition, the presence of non-tensioned steel affects the

deformational behavior of a prestressed concrete beam after the

initial cracking (49). It was concluded by Shaikh and Branson (49),

that the~ deflection in a beam with non-tensioned steel as compared

to the deflection of an identical beam without non-tensioned steel may

be greater, comparable, or considerably smaller depending on

whether the applied transverse load is approxim.ately equal to, some

what greater than or considerably greater than the cracking load.

Failure of the beam is usually the res ult of failure of the

compressed concrete. However, a beam with a very small percentage

of reinforcement may fail by fracture of the reinforcement. The third

stage, however, is not exhibited by beams having a high value of

steel percentage. The first two stages described above can be seen

clearly for the laboratory beams in Figures 32 to 34 (the steel percent

age varied from 0. 93% to O. 38% for rectangular beams and was of the

order of O. 1 % for the composite beams).

106

2. The level of prestress affects the shape of the load-deflection

curves. An increase in the level of prestress tends to increase the

load required to produce the flexural cracking and thus extends the

first stage. For example, Beam Al (whose prestress level is

greater than that of either Beam A2 or Beam A3) has a cracking load

of 3. 57k as compared to 3. 04k for Beam A2 and 2. 67k for Beam A3

(see Figure 32 and Table 5).

3. It is observed that for most of the beams (8 out of 9) studied

under single load cycle (see Table 5 ), the computed values of deflection

are smaller than the observed values of deflection. It is also observed

that the discrepancy between the computed and measured deflection

curves increases as the applied transverse load approaches the ulti

mate load capacity of the beam. Realizing that the tendency of con

crete to creep under load exists even for very rapid rates of loading

(52), it may reasonably be assumed that the discrepancy between the

computed and observed deflection curves is due to the creep of con

crete, Each load cycle required about 45-60 minutes to complete,

This creep effect has not been accounted for in the development of

Eq. (38). No attempt, however, is made to modify Eq, (38) for

creep effects, because the use of Eq, (38) gives reasonable estimates

of deflection (from a design point of view) up to 1, 5 to 2. 0 times the

working load.

107

4. The use of Eq. (38) resulted in computed deflections being

slightly greater than the observed deflections in most of the beams

(8 out of 10) in Reference (49), while in the current study the use of

the same equation results in the computed deflections being slightly

smaller than the observed deflections. This effect appears to be due

to the presence of non-tensioned steel in the beams reported in Refer

ence (49) which tends to reduce the creep effect and to further distri

bute the cracks along the beam.

5. The composite Beams BZ and B3 exhibit greater resistance

to applied loads than non-composite Beam Bl due to the inherent in

creased stiffness of the former (see Figure 33 and Table 5).

6. There does not seem to be any significant difference in the

load-deflection response of composite beams for which slabs have

been cast at different times. Both Beams BZ and B3 have almost

identical load-deflection curves (see Figure 33). However, there

could be a significant difference in the ~deflections (when referred

to the position before prestressing) due to the difference in the time

dependent contribution to camber (see discussion in Chapter 3).

5. 3 Repeated Load Tests of Pres tressed Members

Under single cycle loading, the load-deflection response of

prestressed concrete members can be reasonably predicted in both

the 1uncracked' and 'cracked' stage. This has been discussed in

108

Section 5. 2. However, under repeated loading, the 'load-deflection'

response is different.

To understand clearly the effect of repeated loads on pre

stressed concrete beams, it is necessary to know the effect of repeated

loads on the two components of pres tressed concrete, i.e., plain con

crete and pres tressing steel. Shah and Winter (22_) studied the behavi

or of plain concrete prisms with flared ends under uniaxial compres -

sion, cycled at stress levels below the ultimate strength of the prism.

They found that concrete possessed a shakedown limit at around 88 to

95 percent of the ultimate load. Below this level, concrete is rela

tively insensitive to several cycles of loading. Neither the strength

nor the strain capacity is affected below the shakedown limit. Pre

stressing steel like reinforcing steel, behaves (for all practical pur

poses) like elasto-plastic material. Repeated loading at load levels

below the yield strength of the material results in full recovery,

while above the yield strength of the material results in an 'inelastic'

set.

In this study of prestressed concrete beams, it is assumed

that under repeated loading, the stress in concrete is below its

'shakedown limit' and the stress in steel is below the 'yield strength'

of the steel. This implies that (1) if_ the concrete stress at the

repeated load level is· below the shakedown limit and (2) if the steel

109

The determination of deflections in the uncracked region (OA

in Figure 35} and cracked region (ABC in Figure 35) has been dis ...

cussed in Section 5. 2, The use of Eq. (38) implies the determination

of the point Bon the assumption that the slope of OB is proportional

to the effective moment of inertia, Ieff' The reliability of this equa

tion has been accepted (~Q_)(~ __ !). Unloading from the point B along BD

(a line parallel to OA) indicates that there is only elastic recovery.

This is true if the beam is severely cracked. If, however, the beam

is not severely cracked a certain number of cracks will close on

unloading (especially in regions of moments close to the cracking

moment). This will result in a small amount of 'inelastic' recovery.

This is indicated by FD in Figure 35. It follows, therefore, that the

total recovery (FE in Figure 35} is a function of the cycling load ......

the closer the cycling load is to the cracking load, the greater will

be the total recovery. This is also a logical extension of the fact

that when the beam is completely uncracked, the total recovery

(indicated by EF in Figure 35) is equal to the total deflection.

On the basis of the above discussion, the following relation-

ship is suggested for computing the average effective moment of

inertia under repeated loads:

= (40}

where:

110

stress (in the same concrete member at the same load) is below the

yield strength of the steel, then the reloading curve after attaining

the magnitude of the repeated load will follow the single cycle load-

deflection curve as if nothing else had happened (see Figure 35). In

Figure 35, this is indicated by the fact that if OAC is the single cycle

load-deflection curve, and if cycling is done at a load corresponding

to OB', the reloading curve (FB) will reach the point B and will follow

BC as if nothing else had happened.

B'

0 F D E

Deflection, 6

c

Slope of OB is proportional to 1eff

Slope of BF is proportional to I rep

Slope of BD is proportional to

lg

Figure 35 Details of deflections under repeated loadings

111

I is used to compute the recovery during the unloading rep

part of the cycle. (Note that the slope of FB is proportional

to I in Figure 35,} rep

= (40-a)

Ieff = effective moment of inertia as defined by Eq. (38)

lg = gross moment of inertia

Pult = estimated ultimate load based on current ACI procedures in the code

p er = load at initial cracking corresponding to Mer

(using Eq. (35)).

Prep= cycling load or maximum load in a given cycle.

Eq. (40) is valid only if the loading cycle produces cracking,

i e P ) P The value of *l requires some explanation. • · ' rep er·

From Figure 35, it is clear that the slope of the line BF is greater

than the slope of the line OB, but smaller than the slope of the line

BD. Also, the slopes of.lines OB and BD are proportional to Ieff

and lg respectively. For a severely cracked beam (Prep; Pult),

the total recovery consists of only the elastic part of the deflection

corresponding to the magnitude of the repeated load, Prep' For an

uncracked beam (Prep; Per)' the total recovery is equal to the

deflection corresponding to the magnitude of the repeated load, Prep•

The value of * 1 interpolates linearly between the two limits described

above. For example:

112

(a) when Prep = Per' '1Ji = 1, and !rep = Ieff = lg. This is a

(b)

condition of total recovery.

when P ' P , ~ 1 varies between 1 and zero, and !rep is rep/ er

between Ieff and lg• This a condition of some inelastic

recovery due to the cracks being closed.

(c) when Prep = P ult' W 1 = 0, and !rep = lg. This is a condition

of no inelastic recovery due to the cracks being closed. This

may also be considered as a condition of maximum residual

deflection.

Thus, the use of Eq. (40) enables one to predict the effective

moment of inertia under repeated cycles for any given range of

loading.

Also, the use of Eq. (40) in determining the effective moment

of inertia under repeated loading allows the slope of BF (see Figure

35) to become proportional to !rep·

In the development of the relationship in Eq. (40), the follow-

ing are implicitly assumed:

1. Absence of hysterisis loop in the unloading-reloading

sequence.

2. Absence of time-dependent effects due to creep during

the test.

The first assumption is justified on the basis that the stresses

113

due to repeated loading in concrete and steel are well below the shake-

down limit and the yield strength of the concrete and steel respectively.

This has also been observed in the study of reinforced concrete beams

under repeated loading in similar loading regimes (54). The second

assumption is probably justified on the basis of the small time involved

in the tests (see Table 6).

In this work, repeated loads mean a small number of cycles

at loads ranging from 1. 05 to 1. 43 times the working load (this cor-

responds to 55 to 72% of the ultimate load}. The working load is

defined herein as the load at which flexural cracking is initiated. The

following sample calculations indicate the use of Eq. (40) in the deter-

mination of deflections of prestressed concrete members under re-

peated loading.

Sample calculations for the deflection of a prestressed concrete beam under three cycles of loading

To illustrate the procedure outlines above, the midspan deflec-

tion of beam E 1 is computed under three cycles of repeated transverse

loads of the following magnitude:

(1) Prep = 5. 0 kips in the first cycle

(2) Prep = 5.5 kips in the second cycle

(3} Prep = 6. 0 kips in the third cycle. Note that P rep

corresponds to the maximum load in a specific load cycle.

114

The equations needed for the computations are Eqs. (35), (38),

(40), and (41). The pertinent geometrical and material properties

for the example beam are shown in Tables Al-AZ and A6.

For a simply supported beam under a two-point symmetrical

loading system (see Figure 30), the midspan deflection, A, is given

as:

= pa (8a2 + 12ab + 3b2 ) 48 EI

(41)

where: b = distance between the loads

a = distance of each load from the near support

P = total load on the beam

E = elasticity modulus of concrete

I = moment of inertia.

Referring to Figure 30, a = 5. 5 ft; and b = 4. 0 ft. For purposes of

illustration, the computed deflections will be referred to Figure 36.

Deflection, A

Figure 36 Sample Calculations

OABE, EBCF, and FCDG repre-

sent the first, second, and third

cycle respectively. The values of

Prep correspond to OB', OC 1, and

OD' during the first, second, and

third cycle respectively. OA'

represents the 'cracking load'

(also referred to as the working

115

load) and is defined as the load at which flexural cracking is initiated.

Parameters and terms for beam El

Span= 15'; e (midspan}= e (end}= l. 75"; F = 38, 7 kips (deter-

mined as Fi - ti Ft' where Ft is obtained using Eq, (17) in Chapter 4};

As = 0, 3196 in2

; Ag= 48. 0 in2 ; Ig = 256 in4 ; Icr (using Eq. (39}}

= 51. 96 in4 ; Ee (using Eq, (6)) = 3. 34 x 106 psi; Mer (Using Eq, (35))

I = 150. 7 inkips; MDL= 13. 7 inkips; fcb = 490 psi (see Table A6};

f~ = 5680 psi (see Table A6}.

Deflections during the first cycle corresponding to Prep= 5, 0 kips

(OH in Figure 36)

Mmax= Mtransverse load+ Mdead load= 5 x 5. 5 x 12/2 + 13. 7

Ieff (using Eq. (38))

{l (using Eq. (41))

= 178. 7 inkips.

= 174.29 in4

= O. 9422 in

as compared to the observed value of O. 938 in (see Figure 40),

For the unloading stage of the first cycle:

(corresponding to

Mer = 150. 7 inkips}

(based on ultimate equations given in ACI 318-63 code}

h (using Eq. (40a}}

Irep (using Eq. (40))

= 4. 15 kips

= 8. 54 kips

= o. 805

= 190.0in4

Recovered deflection (using Eq. (41)) = 0. 865 in (indicated by HE in Figure 36) with

I = Irep

116

Computed residual deflection = Total deflection - Recovered deflection

= 0, 0772 in

as compared to the observed value of 0, 0710 in (see Table 6),

For loads less than Prep during the unloading stage, the com-

puted deflections are in a linear relationship with the applied trans -

verse load (indicated by BE in Figure 36).

Recovery ratio = Recovered deflection/Total deflection

= o.8650 /o. 9422 = 91. 6%

The reloading curve for the second cycle is the same as the

unloading curve for the first cycle (indicated by EB in Figure 36).

Deflections during the second and third cycles

The computation of recovered deflection in the second and

third cycles is similar to that indicated for the first cycle. Only the

computed results are indicated below:

Cycle prep Deflection in inches No (kips) Total Recovered Residual Recovery Ratio

2 5. 5 1. 2 388 1.005 0.2338 81%

3 6. 0 1. 5697 1.089 0.4807 70%

Comments:

1. The recovery ratio reduces with increasing load. This

was the basic premise on which Eq. (40) was developed.

2. The residual deflection increases with increasing load.

117

This is a direct consequence of increased flexural cracks that remain

open even after unloading is completed.

3. The loading and unloading curves for a given load level

are linear, provided the stress in steel is below the yield strength

of the steel and the stress in concrete is below the shakedown limit

of the concrete. The linearity of the loading and unloading curves for

a given load level has been observed for reinforced concrete beams

also (54).

Repeated load tests (three cycles of loading) were conducted

with a constant load cycle on beams of Group C and with an increasing

load cycle on beams of Group E. Midspan deflections on all the test

beams of these groups were obtained up to loads ranging from 76 to

87 percent of the ultimate loads. The range of the cycling loads

varied from 55 to 72 percent of the ultimate load. Eq. (40) was used

to determine the effective moment of inertia in the computation of

deflections. Eq. (35) was used to determine Mer' and Eq. (39) was

used for the determination of Icr· I

The modulus of rupture, fcb• was

obtained by bending tests on plain concrete specimens for the test

beams.


curves are shown in Figures 37 to 41. Figure 42 shows the variation

between the total deflection (corresponding to the maximum value of

a .... "' .: ....

4.0 ~--,

2.0

1. 0

0 0 o. 1

8.0

'"Cl 7. 0

"' 0 ~

'"Cl

118

Maximum value of repeated load, i.e. k

p = 4. 0 r

Beam Cl

Cycle 2

?

0.2 o. 3 0.4 0.5 0 o. 1 0.2 o. 3 0.4 0.5

Beam Cl -"' >- -Computed (loading & unloading)

Observed (loading &

Loading

Q) 6.0 ... .... - Unloading P. P.. Pr = 4. ok ~ s.01---=+-.-_:__-.-~~-.....L...---'--..L_~--'---,___.....J....._---1

4.0 Bm Type Crack Ld. Ult. Ld.

Comp ,....... '~ Meas ,,omp Meas

Cl Re ct 3. 6!- 3. 7cf 7. 98k 8. 20k

':'For details of computation, refer to text.

OLL---'---L_---1...--'-----'---~---'----'----'--~ 0 o. 3 o. 6 0.9 1. 2 1. 5 1.8 2.1 2.4 2.7 3.0


Figure 37 Observed and computed midspan deflection vs load curve of beam Cl under 3 cycles of repeated loading (one noncomposite prestressed beam)

119

Maximum value of repeated load, P k

::: 7. 0 8.0 .-~./-,~~-.-~~~~~~~~.--~---.~~r'-----,-----~--,~~-.-~~-,

"' P< ] 14. 0 ~ ·~

'O <ti lZ. 0 0

'O

"' ~ 10. 0 §:

<t: 8. 0

6. 0

4.0

z. 0

0

0 • 05 • 10 • 15 .zo .Z5 .30

' ' '

----- Computed (loading and unloading)

Observed (loading and unloading) Beam CZ

6 Loading \.... ----A Unloading -r ---~ ~

Pr = 7. ok ~ - -« 1----··· .. Crack. loaJ Ulti. load

Bm Type .,. ~r:

I Cornn Meas Cornn Meas

k 6.6ok

k k

~ CZ T Bm 6.6Z 11. 83 lZ. }(

.I

/cycle 3

':'For details of computation, refer to text.

' ' 0 0.1 0.2 0.3 0.4 0.5 o. 6 o. 7 0.8 0.9 1. 0

Mids pan deflection in inches

Figure 38 Observed and computed midspan deflection versus load curve of beam C2 under 3 cycles of repeated loading (one non-composite prestressed beam)

Ill p. .....

.... 14. 0 i:: .....

'O ~ 12. 0 -'O Q) ..... P.. 10. 0 p.

<t;

8.0

6.0

4.0

2.0

0

0

120

Maximum value of repeated load, Pr k

= 7. 0

• 05 • 10 • 15 • 2 0 • 25 0 • 05 . 10 • 15

-- - - -- Computed (loading & unloading)

.20 .25 • 30

Beam C3 \

Observed (loading & unloading) _\ 6 Loading --... Unloading ----. /\ 7.0k

- . " ~r -· .. .,... -· -

U. - L...> _ _....... -r Crack. loa< Ulti. load

Cycle I Bm Type ~~ ·'·

3 ,......omn Me~s '""'om~ Meas

6.58k 6.6ok k

2. ii!< l C3 T Bm 11. 83

J '"For details of computation, refer

I to text

. '

0 • 1 • 2 • 3 .4 • 5 . 6 • 7 • 8 • 9 1. 0 Midspan deflection in inches

Figure 39 Observed and computed midspan deflection versus load curve of beam C3 under 3 cycles of repeated loading (one composite prestressed beam)

8. 0 Maximum value of repeated load, Pr= 5. ok, 5. 5k and 6. ok

6. 01---~

"Cl cd 0 10. 0 ~

8.0

6.0

4.0

2.0

0

• 2

----

6

•

0 0.3

• 4 0

. 6

. 2

Computed

in cycles 1, 2, and 3, respectively

• 8 • 4

1. 0 . 6 0

. 8 1. 0 2 4

(loading and unloading)

1. 2 1. 4 6 R 1 0

Observed (loading and unloading) Loading _/.'::,, --

f- )- ~-Unloading ~--

- --r- -

;;/ IF

Crack. loac Ulti. Bm Type

Com;' rorn~ Meas

/ El Re ct 4. 15k 4. 2rf 8. 55k

v " • 6 . 9 1. 2 1.5 1.8 2.1 2.4 2.7 3.0


1 ? 1 4 1. 6 1. 8

---

load ''For details of compu-

Meas tation,

8. 3 lk refer to text

3.3 3.6 3.9 4.2

Figure 40 Observed and computed midspan deflection versus load curve of beam El under 3 cycles of repeated loading (one non-composite prestressed beam)

0

20.0

16.0

12.0

8.0

4.0

0

Maximum value of repeated load, Pr= 10. Ok, IO. 5k & 11. Ok in cycles 1, 2 and 3, respectively for Beams E2 and E3

. 03 . 06 . 09 . 12 . 15 .18 .21 .24 .27 0

' '

-----Computed (loading & unloading) -- Observed (loading & unloading)

6. Bm E2 (loading) • Bm E3 (unloading) - -A Bm E2 (unloading) - ,,--- -0 Bm E3 (loading /\ -

Type Crack. loa• Ulti.

Bm ;:~

• 30 . 15

. 33

. 30 0

. 45 15 .

. 60 • 75 30 45 .

--- <"" - - Beams E2 & E3

"

load ':'For details of compu-refer to text

l?"r "' ,;, tation, Ir<~ -- .,. ...... Meos o~~ Meas

/ 9.49k 9. 6rf I I

Ez~~,:~ T Bm 8.74 9. 10 k

9.60k I 1

E 3 ;:~~:~ T Bm 9.47 8.75 8.65

/

• 90 60

.

0 . 15 . 30 .45 .60 .75 .90 1.05 1.20 1.35 1.50 1.65 l.80 1.95 2.10 2.25


Figure 41 Observed and computed midspan deflection versus load curves for beams E2 and E3 under 3 cycles of repeated loading (two composite prestressed beams)

':":'Only one curve is shown for the computed and measured values (for both beams E2 and E3) because only very small differences in deflection existed between the two beams.

-N N

123

3. 0 ' ., " " u

p.. p.. II II

'O 'O

"' "' 0 0 - -@ @ . 4-< 4-< ., ., 'O 'O

'O 'O ., ., > > " " ., ., "'

., 1. 0

& Cl (Rect) C!!IE 3 ( T Bm) ( 1 19 - 1 4 3 P )

O C2 (T Bm) • 1 • er

~-·> e C3 (T Bm)

o E 1 (Rect) ...... • E2 (T Bm) ..-.- _>-··_,.

-··- C!!I --- 1, 04-1, 15 P~~) .......,; Ill.,.--- _....d: 0 8 pc r -.. .. ··1 1-• - -

I

2. 0

..0 ..0 0 0 1.06 per - -"' "' ...., ...., 0 0

[-< [-< 0

0 1 2 3 4 No. of cycles

Figure 42 Effect of repeated loading (in the cracked range) on total deflections of laboratory beams of Groups C and E

b

TABLE 6

aDETAILS OF REPEATED LOAD CYCLES AND DISCREPANCY IN THE OBSERVED AND COMPUTED VALUES OF MIDSPAN DEFLECTION

FOR BEAMS OF GRPS C & E

c d e Comp Meas Work Load Cycling Ld, Comp. res. def. Meas. res. def.

Detail ult, ult. load, factoi p pr (total)@ end of (total)@ end of ld, ld, PW Pu/

max for cvcles cvcles cycles

Pu Pum Pw 1 2 3 1 2 3 1

u Cl 7.98 fl, zo 3.67 2, 18 7.00 4.5 4.5 4.5 . uo83 • Ob83 • 0685 • 0600

P< CZ 11.83 12. 10 6.62 1, 79 10.25 7.0 7.0 7. 0 • 0046 • 0046 • 0046 • 0040 .... CJ

C3 11. 83 12. 15 6.58 1, 81 9.20 7. 0 7.0 7.0 • 0043 • 0043 • 0043 • 0040

El 8.55 8.31 4. 15 2. 06 7.20 5,0 5.5 6,0 • 0772 • 2338 • 4807 • 0710 [ii

P< E2 18.74 19. 10 9.49 1, 98 15. 60 10, 0 10. 5 11. 0 • 0040 • 0101 . 0267 • 0030 ....

CJ E3 18,76 18,65 9.47 1, 98 14.80 10, 0 10. 5 11. 0 • 0040 • 0101 • 0267 • 0030

a All loads are expressed in kips and all deflections are expressed in inches. b

See Footnote 1, Table 5, c

See Footnote 2, Table 5. d

See Footnote 3, Table 5,

2 , uo I 0

• 0040

• 0040

. 2310

• 0090

• 0080

3 • uo!O

. 0040

• 0040

.480

• 025

• 026

f Worst disc rep. in defl. curves

+ 14o/o

-18%

-34%

+10%

-33%

-33%

eThe magnitudes of residual deflections being very small, any meaningful interpretation on the basis of a percentage of deflection at working load, (say Pw) is difficult. See Sample Calculations also.

f The discrepancy in the deflection curves refers to the load-deflection curves after the cycling loads have been completed, The high values of discrepancy in this column corresponds to about 80-82% of the ultimate load. Also, see Footnote 5, Table 5.

125

the repeated load in a specific cycle, P ) and the number of cycles rep

at various load levels. Table 6 shows the computed and measured

values of ultimate load as well as the computed and measured mag-

nitudes of residual deflection.

Based on Figures 37 to 42, Table 6, and the sample calculation,

the following observations are made:

1. The residual deflection at the completion of a cycle is a func-

tion of the load at which the cycling is done. At cycling loads close to

the ultimate load, the residual deflection is larger than at cycling

loads close to the cracking load (see Table 6 and Figures 37 to 41).

2. Repeated cycles (up to three cycles) of loading at a given load

level does not increase the magnitude of the residual deflection.

Similar observations have been made on reinforced concrete beams

(54) under load levels below the yield strength of the reinforcement.

3. The magnitude of the total recovery decreases with increasing

load (see sample calculations). This was the basic premise on which

Eq. (40) was developed, and is confirmed by observations (see Table

6).

4. The residual deflection at the end of a cycle is also a function

of the geometric properties of the section. This, though obvious, is

clearly seen in Figures 37 and 38, where the .composite beams have

less residual deflection than non-composite beams even at the same

level of loading.

126

5. There does not seem to be any significant difference in the load

deflection response of composite beams under repeated loading for

which slabs have been cast at different times. Both beams E2 and E3

have similar magnitudes of residual deflections and ultimate loads

(see Figure 41 and Table 6).

6. It may safely be concluded, that the relationship suggested by

Eq. (40) gives reasonable agreement between observed and computed

values of deflections provided the stress under repeated loading in

concrete and steel are below the shakedown limit of the concrete and

the yield strength of the steel respectively. In the case of the labora

tory beams, the range of the repeated load varied between 55 to 72%

of the ultimate load. This corresponds to 1. 05 to 1. 43 times the

working load. It is reasonable to expect that as the repeated load

approaches the working load, the total recovery approaches the total

deflection. This has been discussed in detail elsewhere. Also, com

parison with data in the literature confirms the use of Eq. (40) as a

reasonable means of estimating the effective moment of inertia under

repeated loads for reinforced concrete beams under similar loading

regimes (see Section 5. 5 ).

7. It is reasonable to expect that at repeated loads close to the

ultimate load (yield of steel reinforcement in the case of under-rein

forced beams), there will be greater residual deflection (than when

127

steel has not yielded) as well as a hysterisis loop during the loading-

unloading sequence. A detailed study of reinforced concrete beams

in this loading regime has been reported by Ruiz (55).

5. 4 Increasing Load Plus 24-Hour Sustained Load Tests

Although much work has been reported on the effect of sus -

tained load on reinforced concrete beams (i_), (2.Q_) most of these works

referred to beams at early loading ages. In this study, the beams

were loaded (at beam age 6 months) into the 'cracked' or 'inelastic'

range and left in that position for 24 hours.

Increasing load plus 24-hour sustained load tests in the

cracked range were conducted on beams of Group F (one non-composite

and two composite members). Midspan deflections on all the test

beams were obtained up to loads ranging from 79 to 92 percent of the

ultimate loads. The sustained loads ranged from 33 to 92 percent of

the ultimate loads. Eq. (38) was used to determine the effective mo-

ment of inertia, Eq. (35) was used to determine the value of Mer

I

(with Ft= 0). The modulus of rupture, fcb' was obtained by bending

tests on plain concrete specimens of the test beams.

The creep coefficients for computational purposes were based

on information and test results presented in Chapter 3. The following

sample calculations indicate the use of Eqs. (7), (38) and the appropriate

creep coefficients in the computation of deflections of reinforced

128

concrete members under increasing load plus 24-hour sustained

loading.

Sample calculations for the deflection of a reinforced concrete beam under 24-hour sustained loads

Beam Fl is selected for illustrating the calculation of deflection

under 24-hour sustained load in the 'inelastic' range of the load-deflec-

tion curve.

Parameters and terms for Beam Fl:

Span = 15 ft; e (midspan) = e (end) = 2 in; A = O. 6 in2; Ag = 48. 0

in2

; Ig = 256 in4 ; Icr (using Eq. (39)) = 102.6 in4 ; Ee (using Eq. (6))

= 2. 98 x 106

psi; MDL= 13. 7 inkips; Mer (using Eq. (35)) = 27. 5 inkips;

I I

fcb = 430 psi (see Table A6); fc = 4540 psi (see Table A6); Pult (based

on ultimate equations given in ACI 318-63 Code) = 3. 56 k; age of beam

at load deflection test= 201 days

Deflection under sustained load, Psust = l. 2 kips

~ax= MDL+ Mtransverse load

Ieff (using Eq. {38))

lli (using Eq. (41))

= 13. 1 + 1. 2 x 5. 5 x 12 I 2

= 53. 3 inkips

124.lin4 =

= O. 356 in

as compared to the observed value of O. 350 in {see Figure 43),

Experimental C (from 7 days at u

40% RH)

Correction factor for 50% RH (using Eq. (12))

= 1. 95

= 0.94

Correction factor for age of loading (using Eq. (10))

Actual Cu

''Experimental value of Ct/Cu at 1 day (based on loading at 9 days)

Actual Ct for 24-hour loading

':":'Deflection due to sustained loading (using Term (2) of Eq. (16))

129

= o. 684

= 1.95 x o. 95 x 0.684 = 1,26

= 1/8

= 1.26 x 1/8 = 0.158

= • 158 x • 356 x • 85 = • 048 in

as compared to the observed value of 0. 053 in (see Table 7)

''The experimental value of Ct/ Cu and not the computed value (based on Eq. (7)) is used in the calculations, because the validity of the latter for extremely short periods is questionable, although the equality of this ratio (Ct/Cul for various loading ages is implicitly assumed in the equations for creep (see Chapter 3).

':":'The effect of shrinkage is very small (due to the very late age of loading as well as the short time period of the test) and is considered negligible.


curves are shown in Figure 43. Table 7 shows the computed and

measured values of the ultimate load, as well as the computed and

measured values of the deflections due to the 24-hour sustained load.

Based on Figure 43 and Table 7, the following observations

are made:

1. The magnitude of the deflection due to sustained loading (24

hours) is a function of the level of the sustained load. Beam F2 has

8

7

6

"' 5

p.. ·r< ..x: s:: 4 ·r<

"CJ <1l 0 .....

3 "CJ ., ·r< ..... p.. p.. 2 <:

1

0 0

----- Computed

--- Observed

6 Beam Fl

A Beam F2

• Beam F3 Denotes deflec. ~~t«

t-----1 due to 24-hr

• 1

sustained load

• 2 . 3 • 4 • 5

Crack Ld. Ult. Ld. Bm Type .. * ,....omO Meas ""omn Meas

k k k k Fl Re ct 0.42 0.40 3.56 3.62

k k 5. 39k

k F2 T Brr 1. 11 1. 00 5.42

F3 T Brr 1. l lk 1. irf 5. 4lk k 5.61

'~For det;;>ils of computation, refer to text

. 6 • 7 • 8 . 9

,.____ -"\ Beam Fl

1. 0 1. 1 1. 2 Midspan deflection in inches

1.3 1.4

Figure 43 Observed and computed values of midspan deflection for beams of Group F under 24-hr sustained loading (one non-composite and two composite reinforced beams)

.... "' 0

Detail

Fl r..,

"' F2

'" 0 F3

TABLE 7

aDETAILS OF INCREASING LOAD PLUS 24-HR SUSTAINED LOAD TESTS WITH REGARD TO WORKING LOADS, ULTIMATE LOADS AND

DEFLECTIONS UNDER THESE LOADS

b e Working Ult. Load Load Sustained Def. due to Sustained Worst load, c Factor Ld. factor load Discrepancy

PW Comp Meas Pu/PW p /P d Meas in def. curves s w Comp

0.42 3.56 3.62 8.5 2.86 • 048 • 053 +10%

1. 11 5.39 5.42 4.9 4.50 . 062 . 065 +5%

1. 11 5.41 5.61 4.9 2.88 • 032 . 032 +5%

a All loads are expressed in kips and all deflections are expressed in inches. The period of the test varied between 15-25 min for each beam prior to the application of the sustained load and between 10-20 min after the end of the sustained load.

b See Footnote 2, Table 5.

c See Footnote 1, Table 5.

dThe creep coefficient was the experimental value of Ct for the 24-hour sustained loading. See Sample Calculations.

e See Footnote 5, Table 5.

132

a higher deflection under sustained load than Beam F3 due to the higher

level of loading due to the higher level of loading in the former (see

Table 7 and Figure 43). The deflections due to creep is approximately

proportional to the applied load.

2. For extremely short periods of sustained loading (24 hours),

the use of Eq, (7) for the determination of creep coefficients in the

computation of deflections due to sustained loads is, perhaps ques-

tionable. The experimental values of Ct/Cu is used in the computa-

tions. However, the experimental value of Ct /Cu (= 1/8) does not

differ very much from the computed value of C /C (using Eq. (7) t u

= 1/11).

3. The use of Eq, (38) for the determination of the effective

moment of inertia of reinforced beams has been suggested for the

1971 ACI Code (!)(50)~ ). It gives reasonable agreement at loads

very close to the ultimate load also (see Figure 43).

5. 5 Results Reported by Others

The observed load-deflection curves reported by Abeles (56),

Warawaruk, Sozen, and Siess (i:_!_l, Shaikh and Branson (49), and Burns

and Siess (54) are compared with the computed values obtained by

using the methods presented in Sections 5. 1 to 5. 4.

133

Results Reported by Abeles (56)

In his investigation, Abeles reported the load-deflection

response of three groups of rectangular prestressed beams (with

different levels of prestress and steel percentage) under various

conditions of single, repeated, and fatigue load cycles. His primary

interest was in the fatigue loading of pres tressed beams. Single and

repeated load cycle tests were conducted on companion specimens to

obtain a basis of reference. Beams of ordinary and lightweight con

crete were included in the study. Of the 16 beams tested, only A01'~

and ALl~' are used for purposes of this study. Table CS shows the

details of the beams used in this study. Beams AOl~' and ALl'~ were

studied under three cycles of repeated loading. However, no measure

ments of residual deflections were reported. Hence no continuous load

deflection curves under repeated load cycles could be plotted and com

pared with the computed results.

Figure 44 shows the comparison between the computed and

observed values of midspan deflection. On the basis of Figure 44, the

following observations are made:

1. Within the working load (the working load being defined as the

load at which flexural cracking is initiated), the use of the gross sec

tion properties along with the computed modulus of elasticity of con

crete (using Eq. (6)) gives excellent agreement between the computed

(A) Data from Reference (56) ~ 1.8 ,-~--,~~--.~~-.--~~.---=:;.,,.---~--, .0 u Bm AOl* ,.-.~ i:::

"""' i::: 1. 5 ·~

~ 0.43 11

0. 49"

0 • 3 • 6 . 9 1.2 1.5 1.8

(B) Data from Reference (!!) 1.2~~~~~~~~~--,~~~~~~--,

1. 0

6 RB34. 126

e RB34. 093

D RB34. 031 .·

• 08 11

RB34.031 .04" OIE::__....L~-1-~---l~~-'-~-'-~~

0 • 2 • 4 • 6 .8 1.0 1.2

Observedcmidspan deflection in inches

Figure 44 Observed and computed midspan deflection (using Eqs. (38) and (41) for beams under static loading as in (A) (Data from Reference 56) and as in (B) (Data from Reference 41)

135

and observed values of midspan deflection,

2. In the cracked stage, the scatter between the computed and

measured deflections is noticeable. The magnitude of this scatter

increases with an increase in the applied load. This is probably due

to the omission of creep effects in the determination of the effective

moment of inertia using Eq, (38) (see Section 5. 2 for discussion).

However, the magnitude of the scatter is within±_20% for loads which

are about 1. 75 times the working load.

3. In the cracked stage, the computed values (using Eq. (38) for

the determination of effective moment of inertia) of midspan deflection

are greater than the observed values of midspan deflection. Similar

results have been observed by the ACI Committee 435 ('.!_)in the study

of reinforced concrete beams containing 'compression' steel. This

is probably due to the fact that the presence of compressive steel

tends to lower th.e neutral axis and thereby retard the formation of

cracks. (For a discuss ion of this phenomenon as related to other

types of prestressed concrete beams, see Section 5,6),

Results Reported by Warawaruk, Sozen, and Siess (i:.!_)

In a comprehensive study of the strength and behavior in

flexure of pres tressed concrete beams, Warawaruk, et al. reported

the load-deflection response of both post-tensioned and pretensioned

beams. A large number of variables were studied, the most important

136

of which were the steel percentage, type of concrete, loading condi

tions, and type of bonding of reinforcement with the concrete. Of the

82 beams tested, only beams RB34, 126, RB34. 093, and RB34. 031

(pretensioned) are used in this study, The details of these beams are

shown in Table C6. The loading was done statically by a symmetrical

two-point loading system.


observed values of midspan deflection. On the basis of Figure 44,


1, Within the working load, the use of the gross section properties

along with the computed modulus of elasticity of concrete gives execl

lent agreement between the computed and observed values of midspan

deflection,,


observed values of midspan deflection is noticeable and the magnitude

of this scatter increases with an increase in the applied load. This is

probably due to the omission of creep effects in the determination of

the effective moment of inertia using Eq, (38) (see Section 5. 2 for

discussion). However, the magnitude of the scatter is within:!::_ 20%

for loads which are about 2, 0 times the working load.

3. The beams studies in this report did not have 'compression'

steel. Also, in the cracked stage, the computed values (using Eq,

13 7

(38) for the determination of effective moment of inertia) of midspan

deflection were smaller than the observed values of midspan deflection.

This is consistent with the results described in Section 5. 1 for the

laboratory beams and is probably due to creep effects that have been

neglected in the development of Eq. (38). (For a discussion of this

phenomenon as related to other types of pres tressed concrete beams,

see Section 5. 6.)

Results Reported by Shaikh and Branson (49)

In a comprehensive study of the effects of non-tensioned steel

on the behavior of prestressed concrete beams, Shaikh and Branson

reported the load-deflection response of 12 pretensioned concrete

beams containing various types and quantity of non-tensioned steel.

The details of these beams are shown in Table C7. The loading was

done statically by a symmetrical two-point loading system.


observed values of midspan deflection. On the basis of Figure 45,


1. Within the working load, the use of the gross section proper

ties along with the reported modulus of elasticity results in excellent

agreement between the computed and observed values of midspan

deflection.

"' QJ

. .i::: u 1:1

•rl

"' "" •rl s "" QJ .... [

3 (A) Data from Reference (49)

0 Bm I B 1 ti Bm IV B 1 II

eBm I B2 l'lBm IV B2 6BmIB3 181BmIVB3 DBm II Bl

2 •BmIIB2 ®BmIIB3 eBmIIIBl <JBmIIIB2 t>BmIIIB3

deflection under working load for all the beams varied between 0. 2 11

-

s 0 o. 4 11

0 0 u 1 2

(B) Data from Reference (54) .6.---~--~-~--~==r-.--~

ABmJ9

. 5 • Bm J 1 O t----+--+-----11-----i

• BmJll

• 3

l>wl • 2

J9 • 02"

• 1 JlO • 02 11

Jll • 04"

0 3 0 • 1 • 2 • 3 • 4 • 5 • 6

Observed midspan deflection in inches

Figure 45 Observed and computed midspan deflection (using Eqs. (38), (40), and (41) for beams under static loading as in (A) (Data from Reference 49) and for beams under repeated loading as in (B) (Data from Reference 54)

-w 00

139


observed values of deflection is noticeable and the magnitude of the

scatter increases with an increase in the applied load. The exclusion

of creep effects in the determination of effective moment of inertia

using Eq. (38) probably causes an underestimation of deflections.

However, the magnitude of the scatter is within ±_20% for loads which

are about 2. 0 times the working load.

3. In the cracked stage, the computed values (using Eq. (38) for

the determination of effective moment of inertia) of midspan deflection

are slightly greater than the observed values of midspan deflection.

This is probably due to the presence of non-tensioned steel which

tends to reduce the creep effect and to further distribute the cracks

along the beam. {For a discussion of this phenomenon as related to

other types of pres tressed concrete beams, see Section 5. 6.)

Results Reported by Burns and Siess {54)

In a detailed study of the effects of repeated loading on the

behavior of reinforced concrete beams, Burns and Siess reported the

load-deflection response of 18 beams. A large number of variables

were studied, the most important of which were the steel percentages,

and the loading regimes. Of the 18 beams tested, only beams J9,

JIO and Jll are included in this study. The beams were unloaded

and reloaded at several stages before and after the yielding of the

140

tension reinforcement. The details of the beams are shown in Table

C8. The loading was done by a symmetrical one-point loading system.

This study indicates that the unloading and reloading from any point up

to the ultimate did not affect the carrying capacity of the beam. The

stiffness of the beam, as measured by the reloading slope of the load

deflection curve was found to depend on the amount of 'inelastic' defor

mation. This is consistent with the results described in Section 5. 2 on

the effects of repeated loading on prestres sed concrete beams.

Figures 45 and 46 show the comparison between the computed

values (using Eq. (40)for the determination of effective moment of in

ertia under repeated loading) and observed values of midspan deflection

under two cycles of loading. The loading stage corresponded to a level

prior to the yielding of the tens ion reinforcement. On the bas is of

Figures 45 and 46, the following observations are made:

1. Within the working load (the working load being defined as the

load at which flexural cracking is initiated), the use of the gross sec

tion properties along with the reported modulus of elasticity of concrete

gives excellent agreement between the computed and observed values of

midspan deflection. The reported and not the computed (using Eq. (6))

modulus of elasticity of concrete because of the large difference that

existed between these two values (a difference of about 20%).

2. At load levels in the cracking range of the beam, the scatter

40

30

"' P< .... "' 20 i::: ....

'O ro 0 10

...:i

0

0

----- Computed

Obs.

=6.60k 7.50k er

k u= 33. 6 35.50k

Beam J9

0.2 0.4 0.6 0.8 0

Comp. k p =3.80

er k

Pu=24. 2

Obs.

3. 9<}<

26 Ok -~""'ield of

Re inf.

Beam J 10

0.2 o.4 o.6 o. 8 Midspan deflection in inches

Observed

Comp.

p =2. 15k er

p =17.25k u

Re inf.

Beam Jll

0 0.2 0.4 0.6 0.8

Figure 46 Comparison of computed and observed values of midspan deflection of beams in Reference (54), under two cycles of repeated loading (three non-composite reinforced beams)

142

begins to be appreciable, and the magnitude of the scatter increases

as the applied load approaches the ultimate load (in this case, the

yielding of the tension reinforcement). However, the scatter is with-

in + 20% for loads which are about 1. 75 times the working load.

Summary of Results Reported by Others

A dimensionless plot (between load and deflection) is also

shown in Figures 47 and 48 for prestressed rectangular and T-beams

(composite or monolithic), respectively. The following observations

are relevant to these figures: (Figures 4 7 and 48)

The allowance of 'severe cracking' in reinforced concrete

beams as compared to 'no cracking' in fully prestressed beams and

'some cracking' (corresponding to the modulus of rupture of concrete)

in partially prestressed beams at service loads, indicates the incon

sistency of the current procedures in the design of reinforced and

prestressed concrete members. One of the reasons for this incon

sistency has been the unavailability of a reliable and simple method

to predict the deflections under 'cracked' conditions for prestressed

concrete members. Figures 4 7 and 48 show the load-deflection

response (on a dimensionless plot) of 24 non-composite prestressed

concrete beams (containing various amounts of tensile, compressive

and non-tensioned reinforcement) and 6 composite prestressed con-

crete beams respectively. Both static and repeated loading resu.lts

are included. Average curves for different steel percentages are

143

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um

1. 0

Figure 47 Range of validity of Eqs. (38)(40) and (41) for rectangular beams with different steel pfOrcentages'~ -- included in this dimensionless plot are also the results from studies made on rectangular prestressed beams from References (.!!_), (49), (54) and (~) as well as the current study.

*The value of p' refers to compressive steEel for curves (]) and © and to nontensioned steel for the other curves,

and p refers to tensile steels for all curves.

6 82

.& 83 A.C2

£C3

o E2

e E3

I

f-l I

144

DESI GM I ZOME -I

.I I

- - -1- - - ----'-I I

I I

----MO p

CD ·II @ ·11

@ '14

© ·14

® ... © ·II

• 1

0

0 I

0 I I

0 I

• 2 • 3 .4 • 5 • 6 • 7 • 8 (Load, P /Measures ultimate load, P )

um

• 9 1. 0

Figure 48 Range of validity of Eqs. (38), (40), and (41) for T beams with different steel percentages* -- included in this dimen-11ic>nless plot ate the results from the current study only.

*The values of p and p' in the figure refer to tensile and compressive steel percentages.

145

also indicated in the figures. The computed values of midspan deflec

tion were based on the methods developed in Sections 5. 1 to 5, 4.

For purposes of discussion, the total load range is divided into

three stages --{i) the 'uncracked stage {O - 30% of the ultimate load),

{ii) the 'cracked' stage or "design zone" (30 - 60% of the ultimate load)

and {iii) the 'severely cracked' stage {60 - 100% of the ultimate load).

The following observations refer directly to these figures.

1. In the 'uncracked' stage of non-composite and composite

beams, the variation between the computed and observed values of

midspan deflection is less than±:_20%. The working load of a fully

prestressed beam usually falls within this stage. This confirms the

use of the gross section properties in the determination of midspan

deflections.

2. In the 'cracked' stage of non-composite and composite

prestressed beams, the variation between the computed and observed

values of midspan deflection is still less than+ZO%. However, the ten

dency for this scatter to increase is noticed in the shape of the average

curves. The working load of a partially prestressed beam usually

falls within this range. This suggests the use of the effective section

properties {using Eq. (38) or {40)) as a reasonable method in the

determination of midspan deflections.

146

3. In the 'severely cracked' stage of non-composite and

composite pres tressed beams, the variation between the computed

and observed values of midspan deflection increases markedly as the

applied load approaches the ultimate load. The working load of a pre

stressed beam is, of course, never within this stage. This suggests

the invalidity of the use of Eq. (38) or (40) in the determination of the

effective moment of inertia in this load range.

4. It is noticed that for prestressed beams (in the 'cracked'

and 'severely cracked' stage) and containing only tensile reinforce

ment, the computed values of midspan deflection tend to be smaller

than the observed values of midspan deflection. This appears to be

due to the omission of 'creep effects' in the determination of deflec

tions using the effective moment of inertia (for range of variation, see

(7) below).

5. It is noticed that for prestressed beams (in the

'cracked' and 'severely cracked' stage) and containing both tensile

and compressive reinforcement, the computed values of deflection

tend to be greater than th_e observed values of midspan deflection.

It is believed that this is due to the presence of compressive reinforce

ment which reduces creep and also lowers the neutral axis, thereby

retarding the formation of cracks. Similar observations have been

reported in the AC! Committee report (i) for reinforced concrete

147

beams containing both tensile and compressive reinforcement. (For

range of variation, see (7) below.)

6. It is noticed that for pres tressed beams (in the

'cracked' and 'severely cracked' stage) and containing tensioned and

non-tensioned steel, the computed deflections differ slightly from the

observed values of midspan deflection. However, the variation

between the computed and observed values of midspan deflection for

these beams are small when compared to the variation


beams containing only tensioned steel. This is probably due to the

presence of non-tensioned reinforcement that tends to reduce the

creep effect and to further distribute the cracks along the beam.

7. One can safely conclude that 'cracking' (corresponding

to concrete stress es greater than the modulus of rupture) can be

allowed in prestressed concrete members provided the deflections

under such loads satisfy the appropriate serviceability requirements.

When compared to the measured deflections, the use of Eq. (38) for

the effective moment of inertia of prestressed concrete members will

result--(i) in smaller deflections (for prestressed beams containing

only tensile steel), (ii) in larger deflections (for prestressed and rein

forced beams containing both compressive and tensile steel), and (iii)

in very slight deviation from the measured values (for prestressed

148

beams containing both tensioned and non-tensioned steel). However,

the scatter between the computed and observed values of midspan

deflection in all the cases studied herein is within ::+:_20% for loads that

range up to 60-70% of the ultimate load. The corresponding load

range for composite beams is of the order of 75-85% of the ultimate

load.

5. 6 Summary and Conclusions

In Sections 5. 1 and 5. 4 of this chapter, methods were pre

sented for the computation of midspan deflections in both the

1uncracked 1 and the 'cracked' stages of prestressed and reinforced

concrete beams under static or repea"ted loading. Comparisons with

observed values of midspan deflection were made with laboratory

beams of this study (Groups A, B, C, D, E, and F) in Section 5. 1 to

5. 4, and with other data from the literature in Section 5. 5.

On the basis of Figures 32 to 48, and Tables 5, 6, and 7 as

well as the specific conclusions in the earlier sections, the following

general observations are made:

1. In the 1uncracked' or 'elastic' range, the use of the gross sec

tion properties along with the computed values of the elasticity modu

lus of concrete (using Eq. (6)) shows excellent agreement between the

computed and observed values of midspan deflection for both rein

forced and prestressed concrete beams under single or repeated

149

load cycles. (See Sections 5. 1 to 5. 3.)

2. The termination of the 'elastic' or 'uncracked' stage (herein

defined as the cracking load or working load for prestres sed members)

can be predicted with confidence for both reinforced and pres tressed

I concrete beams using the modulus of rupture, fcb· (See Figures

32-43.)

3. The allowance of 'severe cracking' in reinforced concrete

beams as compared to 'no cracking' in fully prestressed beams and

'some cracking' (corresponding to the modulus of rupture of concrete)

in partially pres tressed beams at service loads, indicates the inc on-

sistency of the current procedures in the design of reinforced and pre-

stres!!.ed concrete members. One of the reasons for this inconsistency

has been the unavailability of a reliable and simple method to predict

the deflections under 'cracked' conditions for prestressed concrete

members. Figures 47 and48 show the load-deflection response (on

a dimensionless plot) of 24 non-composite prestressed concrete beams

(containing various amounts of tensile, compressive and non-tensioned

reinforcement) and 6 composite prestressed concrete beams respec-

tively. Both static and repeated loading results are included. Average

curves for different steel percentages are also indicated in the figures.

The computed values of mids pan deflection were based on the methods

developed in Section 5. 1 to 5. 4.

150

For purposes of discussion, the total load range is divided into

three stages -- (i) the 'uncracked' stage (0 - 30% of the ultimate load),

(ii) the 'cracked' stage or 'design zone' (30 - 60% of the ultimate load)

and (iii) the 'severely cracked' stage (60 - 100% of the ultimate load),

The following observations refer directly to these figures.

a. In the 'uncracked' stage of non-composite and composite

beams, the variation between the computed and observed values of mid

span deflection is less than±_20%. The working load of a fully pre

stressed beam usually falls within this stage. This confirms the use

of the gross section properties in the determination of midspan

deflections.

b. In the 'cracked' stage of non-composite and composite

prestressed beams, the variation between the computed and observed

values of midspan deflection is still less than±_20%. However, the

tendency for this scatter to increase is noticed in the shape of the

average curves. The working load of a partially pres tressed beam

usually falls within this range. This suggests the use of the effective

section properties (using Eq, (38) or (40)) as a reasonable method in

the determination of midspan deflections.

c. In the 'severely cracked' stage of non-composite and

composite prestressed beams, the variation between the computed

and observed values of midspan deflection increases markedly as the

151

applied load approaches the ultimate load. The working load of a pre

stressed beam is, of course, never within this stage. This suggests

the invalidity of the use of Eq. (38) or (40) in the determination of the

effective moment of inertia in this load range.

d. It is noticed that for prestressed beams (in the 'cracked'

and 'severely cracked' stage) and containing only tensile reinforcement,

the computed values of midspan deflection tend to be smaller than the

observed values of midspan deflection. This appears to be due to the

omission of 'creep effects' in the determination of deflections using

the effective moment of inertia. (For range of variation see (g) below).

e. It is noticed that for prestressed beams (in the 'cracked'

and 'severely cracked' stage) and containing both tensile and com pres -

sive reinforcement, the computed values of deflection tend to be

greater than the observed values of midspan deflection. It is believed

that this is due to the presence of compressive reinforcement which

reduces creep and also lowers the neutral axis, there by retarding the

formation of cracks. Similar observations have been reported in the

ACI Committee Report ('.!_) for reinforced concrete beams containing

both tensile and compressive reinforcement. (For range of variation,

see (g) below.)

f. It is noticed that for prestressed beams (in the

'cracked' and 'severely cracked' stage) containing tensioned and

152

non-tensioned steel, the computed deflections differ slightly from the

observed values of midspan deflection. However, the variation


these beams are small when compared to the variation


beams containing only tensioned steel. This is probably due to the

presence of non-tensioned reinforcement that tends to reduce the

creep effect and to further distribute the cracks along the beam.

g. One can safely conclude that 'cracking' (corresponding

to concrete stresses greater than the modulus of rupture) can be

allowed in prestressed concrete members provided the deflections

under such loads satisfy the appropriate serviceability requirements.

When compared to the measured deflections, the use of Eq. (38)

for the effective moment of inertia of prestressed concrete members

will result -- (i) in smaller deflections (for prestressed beams con

taining only tensile steel), (ii) in larger deflections (for pres tressed

and reinforced beams containing both compressive and tensile steel),

and (iii) in very slight deviation from the measured values (for pre

stressed beams containing both tensioned and non-tensioned steel).

However, the scatter between the computed and observed values of

midspan deflection in all the cases studied herein is within:!:_20% for

loads that range up to 60-70% of the ultimate load. The corresponding

load range for composite beams is of the order of 75-85% of the

153

ultimate load.

4. If the concrete and steel stress during a repeated cycle is

below the shakedown limit of concrete (as defined in Section 5. 2)

and the yield strength of steel respectively, the following observations

are valid:

a. The use of Eq. (40) is a reasonable and simple method

of estimating the average effective moment of inertia of pres tressed

and reinforced concrete beams under repeated loading. (See Figures

37-41, 45, 46.) The use of Eq. (40) estimates the recovery during the

unloading cycle. During the unloading cycle, there is no change in

the slope of the load-deflection relationship.

b. Repeated cycles (up to 3 cycles) of loading at a given

load level does not increase the magnitude of the residual deflection

(see Figures 37-39). It is reasonable to expect that further increase

in the number of cycles will not increase the residual deflection any

more.

c. Repeated cycles (up to 3 cycles) of increasing load level

increases the magnitude of residual deflection (see Figures 40-41).

d. The magnitude of the percentage of the total recovery

decreases with increasing load (see sample calculations in Section

5. 2 ).

5. For reinforced concrete beams under 24-hour sustained

154

cracking load, the following observations are valid:

a. The magnitude of the deflection due to sustained load

is a function of the level of the sustained load - - the higher the mag

nitude of the sustained load, the greater will be the deflection under

the sustained load. The use of experimentally determined creep

coefficients predict satisfactorily the deflection under sustained loads

(see Figure 43 ).

b. The use of Eq. (38) is a reasonable and simple means

of estimating the effective moment of inertia of reinforced concrete

beams. The reliability of this equation is confirmed by the fact that

this has been suggested for the 1971 ACI Code (_2_!_).

6. For all the laboratory beams reported in this study, the use

of the equivalent rectangular stress block for concrete gives reason

able agreement between the computed and observed values of ultimate

strength.

7. There was no significant difference either in the strength or

the load-deflection response between composite sections for which

slabs have been cast at different times. This was true under both

single and repeated loading (see Figures 33, 38, 39, 41, 43 and Tables

5, 6, and 7) cycles.

155

Chapter 6

SUMMARY AND CONCLUSIONS

Presented in this study are the results of a comprehensive in

vestigation of non-composite and composite prestressed and reinforced

structures using different weight concretes. Principal emphasis is

placed on the initial plus time-dependent effects (prestress loss, cam

ber, and deflection), and on the load-deflection response under single

and repeated load cycles (with constant as well as increasing -load

levels) into the cracking range.

Systematic design procedures are described for predicting the

material behavior and structural response. Continuous time functions

are provided for all needed parameters, so that the general equations

readily lend themselves to computer solution. Flow charts are explained

and typical computer outputs are given for loss of prestress, camber, and

load-deflection calculations in Appendix F. A summary of general para

meters is also given in Chapter 4 for hand calculations.

These procedures are verified by comparisons between computed

and experimental results for the data of this project, and for additional

data in the literature. These data include normal weight, sand-lightweight,

and all-lightweight concrete, non-composite and composite reinforced

and prestressed members, and both laboratory specimens and actual

156

structures. Ranges of variation are shown and sample calculations are

included for the procedures presented.

The problem, and the objectives and scope of the investigation

are defined in Chapter 1. This chapter also includes a review of liter

ature. A description of the experimental investigation of this project is

given in Chapter 2.

Systematic procedures are described in Chapter 3 for predicting

strength and elastic properties, creep and shrinkage characteristics of

different weight concretes, types of curing, and types of cement (Eqs.

2 - 13). Standard equations and correction equations for significant

conditions other than "standard" are outlined for design purposes. This

chapter was developed in this project(~) and in Reference (..!_~). Com

parisons between experimental and computed results are shown to be

quite satisfactory for the data of this project (Figures 2 - 7 and B3).

Procedures for predicting the initial plus time-dependent loss of

prestress and camber of prestressed beams and deflection of reinforced

beams are presented in Chapter 4 (Eqs. 14 - 34). Computed results by

these equations, using both experimental material parameters and gen

eral or average parameter.s, are compared with experimental results

for the laboratory beams and sand-lightweight composite bridge of this

project; and with additional data in the literature (Figures 8 - 29 and

Tables I - 4). Separate steel relaxation tests were conducted, and the

contribution of steel relaxation to loss of prestress in beams (as distin

guished from relaxation tests at constant length) is included in a

157

rational manner. It is concluded that the results in Chapter 4 serve to

substantiate the prediction methods described. The approximate equations

may be used for rough calculations only in some cases.

The ultimate loss of prestress for the sand-lightweight concrete

(composite) prestressed bridge girders was 29% to 31% (see Figure 1,

and Tables 1 and 3). It was determined that loss percentages for bridges

under similar conditions using normal weight concrete will normally be

of the order of 25%; and using all-lightweight concrete will normally be

of the order of 35% or higher. Higher losses for the lighter concretes,

for example, are due primarily to the lower modulus of elasticity

(higher elastic strains for a given stress level), and not, necessarily,

to greater creep and shrinkage behavior.

With respect to different slab casting schedules for composite

prestressed and reinforced beams, an earlier slab tends to reduce the

creep curvature by forming an earlier composite section, and also by

reducing differential shrinkage. On the other hand, the creep effect

for the precast beam concrete under the earlier slab loading tends to be

greater. It appears from this study that the net result of these offsetting

effects is beneficial in both prestressed and reinforced beams (earlier slab

reduces pres tress loss, camber, and deflection). It was found in this

study that the beneficial effect of an earlier slab (3 to 4 weeks versus

9 to 10 weeks herein) is relatively small for prestressed beams and

relatively significant in reinforced beams. The decrease in computed

ultimate prestres s loss and camber for the laboratory beams and bridge

girders herein (see Figures 1, ll, 12, 15, 16, 18, and Tables 1, 2)

was negligible for the laboratory beams; and 2% less prestress loss,

158

and 0.10" less midspan camber, for the bridge girders. Only the numer

ical camber, and not the percentage, is meaningful for the bridge gir

ders, because the total camber is near zero due to the heavy deck slab.

The decrease in the ultimate deflection of the laboratory composite beams

was 0.13" or 30% (see Figures 1, 17, and Table 2). The reason for the

difference in the relative effects between prestressed and reinforced

beams has to do with the offsetting effects of prestress and dead load

(including slab dead load) in the one case, and only additive dead load

effects in the case of reinforced beams.

A detailed discussion of the experimental results and conclusions

is also given in Chapter 4.

From the results in Chapter 4, it is concluded that the procedures

presented will normally agree with actual results within _:'.:15% when

using experimentally determined material parameters. The use of the

general or average material parameters herein predicted results that

agreed with actual results in the range of..± 30%. With some knowledge

of the time-dependent behavior of concretes using local aggregates and

under local conditions, it is concluded that one should normally be able

to predict initial plus time-dependent loss of prestress, camber, and

deflection within about _:'.:20%, using these procedures. Some 41 lab

oratory specimens and actual structures were included in Chapter 4.

In the cases compared, it is noted that most of the results are consider

ably better than these limits.

This project is thought to be the first such comprehensive study

of the initial plus time-dependent material behavior and related

159

structural response of both non-composite and composite structures

using different weight concretes.

Developed in Chapter 5 for the first time is a simple and efficient

design method for predicting the entire short-time load-deflection curve

(or a single point, such as at maximum load) under repeated load cycles

into the cracking range for both prestressed and reinforced members.

This method is based on a procedure developed by Branson (50), (.'.!_),

(30), (42) for predicting the deflection of reinforced beams under single

cycle loading and adopted for the 1971 ACI Building Code (22:_), and applied

to prestressed beams under single-cycle loading by Shaikh and Branson

(49). The effects of increasing load levels in subsequent cycles, and of

24-hour sustained loading are also included. Eqs. (35) - (41), the accom

panying descriptions, Figures 32 - 43, Tables 5 - 7, and the correspond

ing sample calculations serve to illustrate these procedures.

The reliability of the procedures described are indicated by com

parisons between computed results and the experimental data of this

project, and with data in the literature (Figures 31 - 34, 37 - 48, and

Tables 5 - 7).

It was found (Figures 37 - 42, and Table 6) that repeated load

cycles (up to 3 cycles in this project) of short duration did not increase

the deflection at a given load level nor the residual deflection after un

loading. However, repeated cycles to increasing load levels did in

crease the residual deflection after unloading, and also increased the

magnitude of the deflection at a given load level when reloaded (Figures

160

40 - 42 and Table 6). Similar results have been shown in Reference

(54). This is attributed to the effect of greater crack development at the

higher loads, and correspondingly greater residual crack effects.

A detailed discussion of the experimental results and conclusions

is also given in Chapter 5.

From the results in Chapter 5, it is concluded that the procedures

presented for predicting load-deflection behavior of reinforced and

prestressed members will normally agree with actual results within

+ 20% for loads as high as 60% to 70% of the ultimate load for non-

composite beams and as high as 75% to 85% for composite beams under

both single and repeated load cycles. This included partially pre stressed

beams l<11aded well into the cracking range. The accuracy is generally

better than~ 20% for normal working load levels. Some 38 non-composite

and composite specimens were included in Chapter 5 (Figures 31 - 34,

37 - 48, and Tables 5 - 7).

With the aid of the material parameter equations presented in

Chapter 3, and the procedures developed in Chapters 4 and 5, the

structural designer can more reliably than in the past predict the initial

plus time-dependent prestress loss, camber, and deflection (including

effects of repeated load cycles) of non-composite and composite rein

forced and prestressed structures of different weight concretes. As a

result of this study, he can also make a better judgement as to the

reliability of his computational procedures and the range of variation to

be expected between computed and actual results, depending primarily

on the degree of care with which the material properties and parameters

(mainly creep and shrinkage) are determined for a given design.

161

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2. Carlson, R. W., "Drying Shrinkage of Concrete as Affected by Many Factors, 11 ASTM, Proceedings, V. 38, Part II, 1938, pp. 419-440.

3. Hveem, F. N., and Tremper, B., "Some Factors Influencing the Shrinkage of Concrete," ACI Journal, Proceedings, V. 53, No. 8, Feb. 1957, pp. 781-802.

4. ACI Committee 435, "Deflections of Reinforced Concrete Flex-ural Members, " ACI Journal, Proceedings, V. 63, No. 6, June 1966, pp. 637-674.

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6. McHenry, Douglas, "A New Aspect of Creep in Concrete and Its Application to Design," ASTM, Proceedings, V. 43, 1943, pp. 1969-1984.

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8. Ross, A. D., "Creep of Concrete Under Variable Stress," ACI Journal, Proceedings, V. 54, No. 9, Mar. 1958, pp. 739-758.

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162

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ACI Journal, Proceedings, V. 66, No. 1, Jan. 1969, pp.

12. Meyers, B. L., and Neville, A. M., "Creep of Concrete: Influencing Factors and Prediction," Symposium on Creep of Concrete, ACI Special Publication No. 9, 1964, pp. 1-33.

13. Pauw, A., and Chai, J. W., "Creep and Creep Recovery for Plain Concrete," Missouri Cooperative Highway Research Programme, Report No. 67-8.

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ACI Journal, Proceedings, V. 63, No. 2, Feb. 1966, pp. 267-290.

15. Jones, T. R.; Hirsch, T. J.; and Stephenson, H. K., "The Physical Properties of Stt"uctural Quality Lightweight AggI"egate Concrete, 11 Texas Transportation Institute, Texas A & M University, College Station, Texas, August 1959, pp. 1-46.

16. ACI Committee 213, 'Guide for Structural Lightweight Aggre-gate Concrete," ACI Journal, Proceedings, V. 64, No. 8, Aug. 1967, pp. 433-470.

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163

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20. Finsterwalder, Ulrick, "Ergenbnisse von Kriech und Schwind-messungen an Spannbetonbauwerken, 11 Beton und Stahlbetonbau (Berlin}, V. 53, No. 5, May 1958, pp. 136-144.

21. Lofroos, W. N., and Ozell, A. M., "The Apparent Modulus of Elasticity of Prestressed Concrete Beams Under Different Stress Levels, " Pres tressed Concrete Institute Journal, V. 4, No. 2, Sept. 1959, pp. 23-47.

22. Mattock, Alan H., "Precast-Prestressed Concrete Bridges; 5. Creep and Shrinkage Studies," Journal, Research and Development Laboratories, Portland Cement Association, V. 3, No. 2, May 1961, pp. 32-66.

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164

29. Scordelis, A. C., Subcommittee Chairman, Branson, D. E., and Sozen, M. A., ''Deflections of Pres tressed Concrete Members, 11 A.CI Committee 435, Subcommittee 5 Report, A.CI Journal, Proceedings, V. 60, No. 12, Dec. 1963, pp. 1697-1728.

30. Branson, D. E., "Design Procedures for Computing Deflec-tions," A.CI Journal, Proceedings, V. 65, No. 9, Sept. 1968, pp. 730-742.

31. Pauw, Adrian, and Breen, J. E., "Field Testing of Two Pre-stressed Concrete Girders," Highway Research Board Bulletin 307, pp. 42-63, 1961.

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35. Abeles, P. W., "Static and Fatigue Tests on Partially Pre-stressed Concrete Constructions, 11 A.CI Journal, .!3:£: ceedings, V. 50, No. 7, Dec. 1954, pp. 361-376.

36. Abeles, P. W., "Partial Pres tressing and Possibilities for Its Practical Application, 11 Prestressed Concrete Institute Journal, V. 4, No. 1, June 1959, pp. 35-51.

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165

38. Abeles, P. W., "Studies of Crack Widths and Deformation Under Sustained and Fatigue Loading," Prestressed Concrete Institute Journal, V. 10, No. 6, Dec. 1965.

39. Burns, N. H., "Moment Curvature Relationships for Partially Prestressed Concrete Beams," Prestressed Concrete Institute Journal, V. 9, No. 1, 1964, pp. 52-63.

40. Hutton, S. G., and Loov, R. E., "Flexural Behavior of Pre-stressed, Partially Prestressed and Reinforced Concrete Beams, "ACI Journal, Proceedings, V. 63, No. 12, Dec. 1966, pp. 1401-1408.

41. Warawaruk, J., Sozen, M. A., and Siess, C. P,, "Strength and Behavior in Flexure of Prestressed Concrete Beams," Engineering Experiment Station Bulletin No. 464, University of Illinois, August 1962, pp. 1-105.

42. Branson, D. E., Subcommittee Chairman, "Prediction of Creep, Shrinkage and Temperature Effects in Concrete Structures," Subcommittee II, ACI Committee 209, Draft Report, April 197 0, pp. 1- 32.

43. Keeton, J. R., "Study of Creep in Concrete, Phases 1-5," Technical Reports Nos. R333-I, II, III, U.S. Naval C. E. Lab., Port Hueneme, Calif., 1965.

44. The California Producers Committee on Volume Change and Affiliated Technical Organizations, "Drying Shrinkage of Concrete," p. 1-40, Mar. 1966.

45. Magura, D. D.; Sozen, M. A.; and Siess, C. P., "A Study of Relaxation in Prestressing Reinforcement," Prestressed Concrete Institute Journal, V. 9, No. 2, Apr. 1964, pp. 13-58.

46. Antill, J. M., "Relaxation Characteristics of Prestressing Tendons," Civil Engineering Transactions, Inst. of Engr., Australia, V. CE 7, No. 2, 1965.

47. Evans, R. H., and Bennet, E. W., Prestressed Concrete, Wiley, New York, 1958.

166

48. Rogers, G. L., "Validity of Certain Assumptions in the Mechanics of Prestressed Concrete, " ACI Journal, Proceedings, V. 49, No. 7, Dec. 1953, pp. 317-330.

49. Shaikh, A. F., and Branson, D. E., "Non-Tensioned Steel in Prestressed Concrete Beams," Prestressed Concrete Institute Journal, V. 15, No. 1, Feb. 1970.

50. Branson, Dan E., "Instantaneous and Time-Dependent Deflec-tions of Simple and Continuous Reinforced Concrete Beams," Part I, Report No. 7, Alabama Highway Research Report, Bureau of Public Roads, Aug. 1963, (1965), pp. 1-78.

51. ACI Committee 318, "Proposed Revisions to the ACI-318-63 Code," ACI Journal, Proceedings, V. 67, No. 2, Feb. 1970, pp. 77-186.

52. Noble, P. M., "The Effect of Aggregate and Other Variables on the Elastic Properties of Concrete," Proceedings, ASTM, V. 31, Part I, 1931, pp. 399-426.

53. Shah, S. P., and Winter, G., "Response of Concrete to Repeated Loads," RILEM International Symposium on the Effects of Repeated Loading on Materials and Structural :E:lements, Sept. 1966, Mexico.

54. Burns, N. H., and Siess, C. P., "Repeated and Reversed Loading in Reinforced Concrete, " ASCE Journal (Structural Division), V. 92, Paper No. 4932, Oct. 1966.

55. Ruiz, M. W., "Effect of Repeated Loads on the Rotation Capacity of Reinforced Concrete Beams," Dissertation, Cornell University, Sept. 1968.

56. Abeles, P. W., Brown, E. I., and Woods, J. 0., "Report on Static and Sustained Loading Test, " Prestressed Concrete Institute Journal, V. 13, No. 4, Aug. 1968, pp. 12-32.

57. Reichart, T. W., "Creep and Drying Shrinkage of Lightweight and Normal-Weight Concretes, " NBS Nomograph 74, National Bureau of Standards, Mar. 1964.

58. Shideler, J. J., "Lightweight Aggregate Concrete for Structural Use," ACI Journal, Proceedings, V. 54, No. 4, pp. 299-328, Oct. 1957.

Ap 1

APPENDIX A

Appendix A includes the details of the laboratory speci

mens and the bridge girders as well as the different

types of concretes used in this project. This also

includes details of the creep and shrinkage specimens

such as the age of loading, ambient relative humidity,

etc.

TABLE Al

. ' ' DETAILS OF LABORATORY BEAMS (GRPS A B C) AND BRIDGE GIRDERS L=86 ',

aAll Beams are 6 11 x 8 11, d=6 11

, Span= 15", bslabs are 20" x 2" 7 11 slab

Beam Groun Group A Group B Group C Bridge

Ream No. Al A2 A3 Bl B2 B3 Cl CZ C3 152-156

c f

Beam D CJ CJ D u u [J u 1f TI Eccentricity in 2.00 2.00 2.00 2.00 2.00 2.00 2,00 2.00 2.00

14.50 6.20

Prestressing . 2-3/8 3-5/16 1-3/8 3-5/16 3-5/16 3-5/16 2-3/8 2-3/8 2-3/8 30-172 ~trand dia in 1-5/16 1-5/16 1-5/16 1-5/16 1-5/16

eA s in2 0.2176 o. 17 34 o. 1377 o. 1734 o. 1734 o. 1734 0.2176 0.2176 0.2176 4.56

p = As/Ag 0.00453 0.00361 0.00287 0.00361 0.00361 0,00361 0.00453 0.00453 0.00453 0.00883

Des. Pre. For, 38.0 30.0 24.0 30.0 30.0 30.0 38.0 38.0 38.0 867.0

Fi' k

Meas. Pre. 37.0 29.6 23,4 30,0 29.9 29.9 38.0 37.9 37.9 867.0

F., kip 1

"1concrete t=+340 t=+ 307 t=+241 t=+310 t=+309 t=+309 t=+ 390 t=+390 t=+390 t= -42 9

Stresses at t= - 107

release of b=-1840 b=-1511 b=-1201 b=-1530 b=-1527 b=-1527 b=-1527 b=-1930 b=-1930 b=-2623

prestress, psi t=-2955

For footnotes see following page,

a o 3/8" Strand, • 5/16" Strand,

Measured stress in all strands Sand Lt. Wt. concrete.

TABLE A 1 (Cont'd)

Measured s tress of bridge girders

in all strands of lab beams = ( 1 72±_ 4) ks i. = 190 ksi. All beams are made of ldealite -

b Six gage WWF, 6" x 6", (As = O. 058 in2 /ft width), slab steel placed in center of slab. No. 3 CT-Stirrups in form of ties for composite slab are spaced at 6 11 c/c in end quarter span and at 22 -1/2" cc in middle half of beam.

c

d

e

Strands placed so that lateral eccentricity is eliminated.

These stresses are computed using the Measured F., t= top fiber stress, b= bottom fiber stress. 1

These initial stresses refer to prestressed section in all cases. The stresses in the case of laboratory beams refer to the end section only. The rectangular (6" x 8") beam dead load, extreme fiber stress at midspan= 218 psi.

The ultimate strength and yield strength (0. 1% offset) were: for the laboratory beam steel 250 ksi and 235 ksi, respectively, and for'the bridge girder steel 270 ksi and 250 ksi, respectively.

f The lower values in this column refer to the center of the girder.

TABLE A2

DETAILS OF LABORATORY BEAMS (GRPS. D, E AND F)

aAll Beams are 6" x 8", d=6", Snan= 15', bslabs are 20" x 3" Beam Group Group D Group E c Group F

Beam No. Dl D2 D3 El E2 E3 Fl F2 F3

Beam [] [] [] [J w w IJ 1d l:J Eccentricity in 1.75 2.00 2.00 1. 75 1. 75 1. 75 2.00 2.00 2.00

Prestressing . 4-3/8 4-5/16 1-1/4 4-3/8 4-3/8 4-3/8 3-1/2 3-172 3-1/2 Strand dia m 3-5/16

e . 2 As 1n o. 3196 0.2312 0.2090 0.3196 0.3196 0.3196 0.6000 o. 6000 0.6000

p =As/Ag c

0.00666 0.00482 0.00435 0.00666 0.00666 0.00666 0.01667 0.01667 0.01667

Des. Pre. For. 56.00 40.60 36.75 56.0 56.0 56.0

F., k 1

Reinforced Meas. Pre.

56. 50 41.00 36.75 56.0 56.2 56.3 ' Fi, kip

. Concrete Beams

d Concrete t=+385 t=+42 l t=+369 t=+375 t=+ 370 t=+370 Stresses at release of b=-2585 B=-2049 b=-1831 b=-2585 b=-2591 b=-2600 pres tress, psi

For footnotes see following page.

TABLE A2 (Cont'd)

a • 3/8" Strand, o 5/16" Strand, a 1/4" Strand, x 1/2" bar, Measured stress in all strands of lab beams = (175 ±.. 2) ksi.

b See Footnote b, Table A 1

c The value of p for reinforced beams is As/bd.

d These stresses are computed using the Measured Fi: t =top fiber stress, b =bottom fiber stress. These initial stresses refer to the prestressed section in all cases. The stress in the case of laboratory beams refer to the end section only. The rectangular (6" x 8") beam dead load, extreme fiber stress at midspan are 178 psi, 208 psi, 208 psi for the beams of Group D, E, and F, respectively.

e See Footnote e, Table A 1

TABLE A3

DETAILS OF CONCRETE MIXES AND MIXING PROCEDURE FOR LT-WT CONCRETES

Cone rete for Description Grps A, B, C& Group D Group E

Bridge Girders Mix design objectives

Cone. Qty, 1 cu vd 1 cu vd 1 cu vd Cone. str. @28d 5000 psi 5000 psi 5000 psi

Mix ingredients Cement (Type I) 705 lbs 752 lbs 705 lbs

F. aggregate Sand - 1395 lbs Haydite agg. (3/16" Sand - 1150 lbs to dust) - 950 lbs

c. aggregate ldealite Agg. (60% Haydite Agg. (3/ 4" Haydite Agg. (3/4" of 3/4 to 5/16 & to 11'4) - 700 lbs to"' 4) - 825 lbs 40% of 5/16 to 4'"8)

822 lbs

Water 35.0 gal 42. 0 gal 42. 0 gal

Dar ex 6. 5 oz 7. 0 oz 6. 5 oz

WRDA 50 oz 53, 5 oz 50 oz

Mixing procedure: 1, Proportion and batch fine aggregate and coarse aggregate. 2, Add 50% of total water requirement. 3. Mix for approximately 2 min. 4. Proportion and batch cement. 5, Add 12. 5% of water requirement.

Group F

1 cu vd 4000 psi

611 lbs

Sand - 1250 lbs

Haydite Agg. (3/4" to #4) - 825 lbs

40. 0 e:al

5. 7 oz

43. 5 oz

6. Add Darex (in solution with 3 gallons of water), WRDA and the remaining water while adjusting to 2-1/2" slump.

TABLE A4

a-gCONCRETE PROPER TIES (GRPS A, B, C AND BRIDGE GIRDERS), TEMPERATURE AND HUMIDITY DA TA

Concrete Batch Gp.A Gp. B Gp. C Slab Slab Slab Slab 1 Bridge

Property SLt. Wt SLt. Wt SLt. Wt BZ CZ B3 C3 Lt. Wt N. Wt N. Wt N. Wt N. Wt

I

fc (7 days) psi 6700 5500 6150 - - - - - - - - 5600

I

f (28 days) psi 9350 8150 c 8750 4800 4140 5100 4300 6100

Unit Wt (Wet) pcf 124.0 124. 0 125.0 -- -- - - - - - -

U, Wt (Dry-7d) pcf 123.0 123. 5 123.5 153 152 152 153 122. 0

Meas. Air Ent. % 4,0 6. 0 6.0 -- -- -- -- - -

Slump in 2,0 2.5 2.5 2.5 2.5 3.0 2.5 - -c

of 3.04 Modulus - - - - a. 3. 20 - - - - - - - - a.

Elasticity ps~ -- -- b. 3.33 - - - - -- - - b. 3. 10

at 7 days x 10 3,68 ~ c. 3,55 - - -- - - - - c. 3. 32

c 3. 28 Modulus of -- -- a. - - - - -- - - --

Elasticity psi - - - - b. 3,58 - - -- - - - - --at 28 days x 106 ~ ~ c. ~ ~ 3.97 !:..!.!. ±:.-22. 3,47

For footnotes, see following page.

!;;Bridge Slab

N. Wt

3500

- -

145

- -

--

- -- -- -

- -- -

3,41

TABLE A4 (Cont'd)

a Lab. temp: 61-85 deg. F., avg. temp. 78 deg. F. Lab. relative humidity: 2.5-61%, avg. rel. hum. 40%. Avg. rel. hum. for central Iowa (from U.S. Weather Bur.): Jan. -79%, July-66%, Mean Annual 71%. For Spr-Sum-Fall, use 70%.

b Stress levels for creep tests were approx. design stresses for lab. beams: I I

Mix Strength, fc, at 7 days Stress Level for Creep Tests % of 7d - fc

Gp. A 6700 psi 2.010 psi 30% Gp. B 5500 1375 2.5 Gp. C 6150 1845 30

C I The modulus of elasticity values are as follows: a. Measured secant (to O. 5 fc) mod. of el.,

d

b. Measured initial tangent mod. of el., c. All values underlined are computed using Ee = 33 J;,;3£~ , psi.

Computed values Girder No.

152. 153 154 155 156

of modulus of elasticity at release for bridge Age at Release Strength at Rel.

2. days 5160 psi 2. 4670 2. 4685 3 5130 3 4440

girders: cMod. of El. at Rel.

3. 19 x 106 psi

.h.Q.! ~ 2.:12. 2..96

e Computed mod. of el. of pres. units at time of slab casting, cEc x 106 psi: Gp. B-·-~· 4. 30; Gp. C--4.2.3, 4.44; Girders 152., 153, 154--3.50; Girders 155, 156--~.

f Concrete specimens for data in this column obtained from casting yard for Bridge Girders 155 and 156. Measurements made in laboratory.

g "Design" values were used for bridge slab concrete.

Property

b I

fc (Release) psi

I

fc (28 days) psi

Unit Wt (Wet pcf

U. Wt (Dry-7d) pcf

Meas. Air Ent, %

Slump in

cModulus of psi

Elasticity 106 at Release x

cModulus of psi

Ehsticity at 28 Days 106 x

TABLE A5

aCONCRETE PROPERTIES (GRPS D, E, & F), TEMPERATURE AND HUMIDITY DATA

Concrete Batch Slab

Gp D Gp E Gp F E2 A. Lt. Wt S. Lt. Wt. S. Lt. Wt N.Wt

4150 4250 3650 --

4925 4950 3950 4200

105,5 122. 2 122.5 15 3. 1

105. 0 122. 0 122.0 15 3, 0

5,5 6.0 5. 0 - -

2.5 3.0 3. 0 --

2.33 2.90 2.70 - --- --

2.52 3. 13 2.80 4,04 -- -- --

For footnotes, see following page.

Slab Slab Slab F2 E3 F3

N. Wt N. Wt N. Wt

- - -- - -

4250 4300 4200

153. 2 154,3 153.5

153,0 154.0 15 3. 0

- - - - - -

- - -- - -

-- - - --

4.06 4. 12 4.04 -- -- --

a

b

TABLE A5 (Cont'd)

Lab. Temp: 60-88 deg. F., avg. temp. 75 deg. F. Lab. relative humidity: 20-65%, avg. rel. hum. 50%.

Stress levels for creep tests were approximate design stresses for lab. beams:

Mix Gp D Gp E Gp F

Age @ release 7 days 9 days

21 days

Strength@ release 4150 psi 4250 psi 3650 psi

Stress level for % initial creep tests

2000 psi 2000 psi 1000 psi

stress 48% 47% 27%

The age at release for Gps D and E refer to the age at release of prestress and for Gp F this refers to the age at which the reinforced beams were in position.

c 131 All values are computed using E = 33 ,./w- fc , psi.

c

..... 0

Ap 11

TABLE A6

CONCRETE PROPER TIES OF LAB BEAMS AT "LOAD-DEF" STUDIES

a

b

c

a

b

c

d

dGrp dGrp Grp Grp dGrp dGrp

Description A B c D E F

I

Computed, fc psi 10850 9350 10050 5650 5680 4540

Measured, r' psi 10560 9420 9995 5600 5725 4600 c

Computed modulus 625 580 600 450 452 405 of rupture, f~b psi

Measured modulus 650 608 628 480 490 430

of rupture, f~ b psi

Computed using Eq. (2). The beams of Group A, B, C, D, E and F were aged 367, 187, 187 .• 187, 189 and 189 days respectively at the time of the load deflection studies.

For lightweight concrete in a drying condition, the modulus of rup-ture ranges from 5Jf';; to llffc. The observed values of the modulus of rupture correspond to approximately 6 ]fr .

Obtained by bending tests on plain concrete members.

The concrete strength of slab concretes of B2, B3, C2, C3, E2, E3, F2, and F3 were 5500, 5760, 4720, 4860, 4200, 4860, 4860 and 4750 psi, respectively.

Ap 12

APPENDIX B

Appendix B includes a discussion of the variables that

affect creep and shrinkage of concretes as well as a

discussion of the correction factors for these variables

with relation to the method developed in the text.

Ap 13

APPENDIX B

Discussion of variables affecting creep and shrinkage (i_)(Q)(~)(42)(~)

Concrete undergoes time-dependent deformations under the

action of sustained loads that are substantially greater than those of

a corresponding unstressed specimen. These additional strains due

to the effect of sustained stress are attributed to creep of the concrete.

Current nomenclature regarding creep of concrete is summarized in

Figure B 1,

When specimens are subjected to uniform axial stress, only

normal strains (both elastic and inelastic) are usually considered.

The elastic strains are stress dependent and recoverable. These

strains include both time-independent and time-dependent strains.

The time-independent elastic strain is also referred to as initial or

instantaneous strain.

The stress independent component of the inelastic strain is

normally called shrinkage. This strain is partially reversible. The

s tress dependent irrecoverable strains include microcracking effects

as well as shrinkage or drying creep resulting from moisture migra

tion due to applied stress. The drying creep cannot be separated

from the irreversible shrinkage.

The total creep strain consists of (a) Basic creep--delayed

strain due to the interaction between solid and fluid phase, (b) Drying

I Elastic Strains I I Inelastic Strains I ' ' • .L • I

Time-Independent Time -De pen - Stress Dependent Stress Independent Strains dent Strains Strains Strains

I I .L .L •

1 • Stress Dependent 1, Stress Dependent 1. Stress Dependent 1. Stress Independent 2. Time-Independent 2. Time-Dependent 2. Time-Dependent 2. Time-Dependent 3. Recoverable 3. Recoverable 3. Irrecoverable 3. Partially Revers.

ee=f 1 (cr) ectl = f2 (cr,t) ect2 = f3 (cr' t) ect3 = f4 (t)

Microcracking Revers.

Basic Creep Creep

Irreversible Shrinkage -----------(Delaved Strain) Drvin' Creep -- Shrinkage (Swell)

I .. . Instantaneous Total 14--- I Shrinkage I

Strain Creep

Total Strain

Fig. Bl Strain Components

Ap 15

creep--consolidation due to seepage of internal moisture, and (c)

Microcracking creep--creep due to irrecoverable creep strains accom

panying microcracking.

The recoverable strains may be time-independent (instantaneous

recovery), time-dependent (delayed strain), or s tress independent

strain recovery (swelling). The independence of creep and shrinkage

of concrete has yet to be established. However, creep and shrinkage

occur simultaneously in concrete structures and, from a practical

standpoint, these may be considered additive in nature. This indepen

dence is assumed through the use of companion stressed and unstressed

specimens, so that the total time-dependent strain minus the free

shrinkage strain is attributed to creep.

The prediction of time -dependent concrete strains is further

complicated by the fact that strains and internal stresses are affected

by the properties of the material as well as by curing and environmen

tal conditions, A comprehensive study of time-dependent concrete

strains includes a large number of variables. These variables are

summarized in Figure B2. A detailed study of all these variables

is beyond the scope of this report. However, with reference to the

principal factors that effect time-dependent concrete strains, the

following are considered in this report in the development of proce

dures for predicting creep and shrinkage:

Parameters affecting Creep and Shrinkage Concrete Strains

1. Min. Memb. Thk. 2. Water-Cement ratio 3. Mix proportions 4. Type of aggregate

3

Material Pro erties

4 5

Mechanical Pro erties

5. Length of curing 6. Curing temp. 7. Curing humidity 8. Environment hum.

6 7 8

Curing

9. Environment temp. 12. No. of load cycle•

10. Time of init. load 13. Unloading period

and time init. shrink- 14. Stress Distr. age considered 15. Stress magnitude

11. Duration of load period 16. Stress rate

9 10 11 12 13

Loading History

14 15 16

Stress Condition

Environmental Conditions

Loading Conditions

------4-.I Time-Dependent Strain Variables

Figure B2. Time-Dependent Strain Variables

~' Parameters studied by Jones (__!2), and used in this report + These numbers refer to the parameters listed above

Ap 17

1, 1 Minimum thickness of member

1. 2 Water-cement ratio in the form of slump and cement content

1, 3 Mix proportions in the form of percent fines and air content

1, 4 Environmental humidity

1, 5 Time of initial loading and time initial shrinkage considered

Presented here is a summary of the principal variables that

affect creep and shrinkage (!.Q), (g), (13), (!.2_), (.!_~)in most cases.

The corresponding nominal correction factors, based on the standard

conditions herein, are given in the text and in Figure B3 (13), (15), - -(!_~). The results in Figure B3, and equations for these curves, were

developed in Reference (.!.§_).

The following comments refer to the nominal correction factors

for creep and shrinkage (from Figure B3 ), which are normally not

excessive and tend to offset each other. For design purposes in most

cases, these (except possibly for the effect of member size as dis-

cussed in the text) may normally be neglected:

Creep correction factors

Slump: C, F. = O. 95 for 2", 1. 00 for 2. 7", 1. 02 for 3", 1. 09 for 4", 1. 16 for 5". Comment--Tends to be offset by effect of member thickness. May be marginal but normally can be neglected.

Cement content (sacks/cu.yd.): C. F. = 1, 00. No correction factor required for concrete of say 5 to 8 sacks per cu. yd. at least.

Percent fines (by wt. ) : C. F. = O. 95 for 3 Oo/o, 1. 00 for 5 0%, 1, 05 for

70%. Comment--Normally negligible.

001.2.-----.--.----71 '" 0 ... u .!'11.01-----±-7""-'-=::J_---i

p., ~ <1J 0 ~ ·.;::: A. Lt, M u u 0. 8(--U-.olL--'--------1

~ 0(17),I,S.Lt,M ~ 57 III, A. Lt, S uO. 6 "57 III N. Wt S

0 2 4 6 a. Slump (in)

0.8

O (15), I, A. Lt, M L:,.(58), I, A. Lt, M

4 6 8 10 b. Cement content

(sacks/cu.yd.)

o(.!i), I, A. Lt,

0 (15), I, A. Lt, 0.8~------~

20 40 60 80 c. Percent fines by

wt. (<1/4 sieve)

o.

4 8 12 16 d. Air content (%)

0 6(58), I, A. Lt,

.8'--.=::...0."-'--'----'--' 6 0(15), I, A. Lt, o. .......-~~---~

jW,5 , I, A. Lt, M

TI , I, S. Lt, M , III, A. Lt,

0 1 , III, N. Wt, • .-.=:.'-"'C.:....-'-......;.-'"'-'

4 6 8 10 40 60 80 f. Cement content

(sacks/cu.yd.)

20 g. Percent fines by

wt. (< 114 sieve)

4 8 12 16 h. Air content (%)

'" 0 ... u

1. 2

<IJ.!'10.9 "" nj ~

.>: 0 ~ .....

•C!.!l· !eshl3ood °C!_!), (E:shh300d

·~ 0 o. 6 - Use for fi5 ~ ,.. 1 yr.

0 o 3 dr in u • '""'"'W'-'~----.....:l 6 12 18 24

Figure B3.

j. Minimum thickness (in)

Nominal creep and shrinkage correction factors for the parameters shown from Ref. 18. Notation: I, III -- type cement; N. wt., S. Lt., A. Lt. Weight concrete; M, S Moist, Steam curing

Ap 19

Air content (in%): C. F. = 1. 00 up to 6%, 1. 09 for 7%, 1. 17 for 8%. Comment--Tends to be offset by effect of member thickness. May be neglected for say up to 7% air.

Minimum thickness of member: C.F. = 1.00 for 6" or less, 0.82 for 12". Comment--Tends to be offset by effect of slumps greater than 3" and air contents greater than 6%. Can normally be neglected for members up to about 10" to 12".

Shrinkage correction factors

Slump: C. F. ::: O. 97 for 2 11, I. 00 for 2. 7 11

, 1. 01 for 3 11, 1. 05 for 4",

1. 09 for 5". Comment--Tends to be offset by effect of member thickness. Normally can be neglected.

Cement content (sacks/cu. yd.): C. F. = O. 87 for 4 sacks, 0. 95 for 6 sacks, 1. 00 for 7. 5 sacks, 1. 09 for 10 sacks. Comment- -Normally negligible for say 5 to 8 sacks per cu. yd. at least.

Percent fines (by wet.): C. F. = O. 86 for 40%, 1. 00 for 50%, 1. 04 for 70%. Comment--May be marginal but normally can be neglected.

Air content (in%): C. F. = O. 98 for 4%, 1. 00 for 6%, 1. 03 for 10%. Comment- -Normally negligible.

Minimum thickness of member: C. F. = 1. 00 for 6" or less, O. 84 for 9". Comment--Tends to be offset by effect of slumps greater than 3". Can normally be neglected for members up to about 8" to 9" minimum thickness.

Ap 20

APPENDIX C

Appendix C includes the details of the test beams from References

(~), (11), (f.1), and (2!_). The loss of prestress and camber of these

beams have been discussed in the text on the basis of the methods

developed therein. Also included are the details of the test beams

from References (Q), (i.2.J, (54), and (56). The load-deflection

response of these beams have been discussed in the text on the basis

of the methods developed therein.

TABLE Cl PROPERTIES OF TEST BEAMS AT UNIVERSITY OF FLORIDA (23)

Cable a I

b Cone. Beam Eccentricitv Concrete Curing Loading A Fa fc rel Stress Profile End Center Type Cond. Age in~ (kip) (psi) End (psi) Center (psi)

1 STRT 2. 16 2.31 Nr wt MC 28d 1. 32 57.9 5030 902 805

2 STRT 2. 02 2. 16 Nr wt MC 28d 1.32 65. 1 5030 982 891

3 STRT 2.20 2.44 Nr wt MC 28d 1. 32 101. 5 5030 1602 1568

4 STRT 1. 91 2.26 Nr wt MC 28d 1. 32 99.9 5030 1457 1460

5 STRT 2,00 2.35 Nr wt MC 28d 1. 32 142. 1 5030 2115 2196

6 STRT 2,03 2.41 Nr wt MC 28d 1. 32 139.5 5030 2108 2184

7 STRT 1. 97 2.22 Nrwt MC 28d 1. 32 93.6 3760 1383 1352

8 STRT 2.30 2,55 Nr wt MC 28d 1.32 87.4 3760 1407 1365

9 STRT 2,33 2.41 Nr wt MC 28d 1. 32 90.0 3760 1461 1354

10 STRT 2.41 2.51 Nr wt MC 28d 1. 32 91.6 3760 1520 1416

a The eccentricities are measured values. bThese stresses refer to the steel cgs, and uses the measured values of F 0 and the net section

properties (+)compression;(-) tension. Remarks: All beams have a span= 19. 5'; all bars are 3/4" rp steel bars.; composite slabs (26" x 3") were cast on beams 1, 4, 6, 8 & 10 at 101, 101, 101, 37 & 93 days after stressing. The steel bars (of Es= 26380 ksi) were not grouted. Beams 1-8 were stored in the lab at 75% humidity & 9-10 were stored in the field at 90% R.H. The mix had a cement content of 6-6. 5 sacks/cu yd of Type I cement.

N

TABLE C2 PROPER TIES OF TEST BEAMS AT UNIVERSITY OF ILLINOIS (24)

c Eccentricitu a I

b Cone, Stress Cable Concrete Curing Loading A Fi fc rel Beam in2 Profile End Center Type Cond, Age (kips) (psi) End (psi) Center (pa i)

~U-1 STR 1. 03" 1. 03" Nr wt MC 5d . 18 26.9 3760 1285 1266

~U-2 STR 1.03" 1. 03" Nr wt MC 5d . 18 26.9 3930 1230 1271

a All eccentricities are measured values.

b These stresses refer to the steel cgs and uses the stress diagram indicated in Reference (24) (+)Compression;(-) Tension.

c All beams were cast of Type III cement with a water-cement ratio of 0. 74-0, 76 and a ratio of (1:2. 98:3, 35) of cement, sand and gravel by wt.

Remarks: All beams have a span= 6 1 , all wires are .196" cp (Es= 30 x 103 ksi). All beams were stored at 50% RH.

!l> 'tl N N

TABLE C3 PROPERTIES OF TEST BEAMS AT TEXAS A & M UNIVERSITY(~)

a b

Cone. Stress As Fi

I

Beam Length Span Cable Eccentricity Cone. Curing Load. fc rel End Center

Profile End Center Type Cond. Age in2 (k) psi {psi) (psi)

Ll-5 40 1 38.16 1 STRT 9. 19 11 9. 19 11 Lt wt SC 2d 1.75 304 4650 1387 1252

L4-5 56 1 54. 29 1 HR PED 7.20 11 9. 60 11 Lt wt SC ld 3.28 564 5540 2116 2322

R 1-5 40 1 38. 16 1 STRT 9. 19 11 9. 1911 Lt wt MC 2d 1. 75 293 4820 1343 1207

R4-5 56 1 54.29' HR PED 5. 82 11 7. 82 11 Lt wt MC 7d 3.93 670 5540 2223 2390

L3-5 56 1 54. 2 9 1 HR PED 5. 55 11 9.05 11 Nrwt MC 2d 3,50 605 5260 2021 2366

a All concrete stresses are computed using Fi and the transferred section properties. {+ compression), (- tens ion) and are at the steel cgs.

b All girders have the section designated as Type B by the Texas Highway Department.

Remarks: All strands are 7/16 11 <p (E8

= 28500 ksi) at an average humidity of 88%. The mix had a cement content of 7 - 7-1/2 s c/cu yd of Type III cement. All harping was at 5 1 from 1:_ of the girder.

TABLE C4 PROPERTIES OF TEST BEAMS AT UNIVERSITY OF MISSOURI (1l_)

Beam Cable Eccentricitu Concrete Curing Load As bF., £~ rel Cone. Stress Profile End Center Type Cond. Age in2 k psi End (psi) Center (psi)

East Para- . 83" 27.55" Nr wt MC 6ld 2.65 452.4 5160 682 1440 Girder bolic

a West Para- . 83" 27.55" Nr wt MC 37d 2.65 450.3 5190 - - - -Girder bolic

a Data from West Girder not available in this reference.

b This force F 0 is after el. losses and is the measured value of the force at the end. The value of F

0 at the center has been estimated from the strain measurements. The steel had an

(Es = 28. 8 x 10 3 ksi).

Remarks: Both girders had a span= 88 1; slab cast at age of concrete of 146d, 54 no of 1/4" cp

strands; 3 diaphrams at 24 1 -10", 49'-6", 74'-2" from end; these are shared at 1/4, 1/2, 3/4 points and stored at 70%.

Ap 25

TABLE CS

aDETAILS OF BEAMS REPORTED BY ABELES(~)

Area b Eff. Ecc. of cModulus Meas. of pres t. prest. of rupture cone.

steel force Ft' steel I

strength Type of fcb As (in2) (in) (psi)

I . Beam cone. (kips) fc (psi)

A01':' Nr wt . 2848 37.2 1, 25 570 5725

ALl'~ Lt wt .2848 32.5 1. 25 486 6600

aThe beams were 4 11 x 9" in section and simply supported on a span of 13' -9". A two point loading symmetrical about the center line of beam (i.e., at a distance of 5' fr om either support) was used for the test.

b The value of the effective pres tr es sing force is based on the reported magnitude of the effective pres tress.

c r.:t The modulus of rupture was based on a value of 6 ,.J fc concrete and 7. 5 fi'c for normal weight concrete.

for lightweight

Remarks: . 0712 in2 of steel area was provided as compressive reinforcement for both beams. The measured steel stress at ultimate was 240 ksi.

Beam

'° N .... • ~

"' µ:i p:;

"' °' 0 . ~

"' µ:i p:;

.... "' 0 . ~

"' µ:i p:;

Area of

Ap 26

TABLE C6

a DETAILS OF BEAMS REPORTED BY WARAWARUK, SOZEN & SIESS (Q)

Eff. Ecc. of Modulus Pre stress Pres tress of

steel, As force, Ft steel rupture (in2 ) (kips) (in)

I

fcb• psi

• 362 40.6 3.08 543

• 211 24. l 3.06 472

• 091 10.8 3,00 544

Meas. cone.

strength I

fc, psi

5230

3970

5280

a The beams were 6 11 x 12" in section and simply supported on a span of 9' -0". A two point loading symmetrical about the center line of the beam (i.e., at a distance of 3'-0" from either support) was used for the test.

TABLE C7

DETAILS OF BEAMS REPORTED BY SHAIKH AND BRANSON (49)

All beams 6" by 8", All d = 6. 5", All span= 15' simply supported

Series No. I II Ill IV

Beam No. 1 2 3 1 2 3 1 2 3 1 2 3

a Actual, Fi (kips) 29. 8 29.0 30. 1 20. 2 20.0 19. 7 30.5 29. 8 29.8 25.2 25.8 24.4

b As (in2 ) • 173 . 173 • 17 3 • 116 . 116 . 116 • 17 3 .240 .240 . 160 • 160 • 16 0

c I (in2 ) As .200 • 400 .600 . 058 • 200 • 400 0 0 . 310 • 080 • 310 .600

d, "'f c in psi 5400 5890 6570 5880

e Modulus of

806 855 I . 894 830 rupture, fcb(ps1)

a The value of the effective pres tress force, Ft was determined as Fi - ii Ft was determined using relationships developed in Reference (il).

b Refers to total tensile reinforcement (tensioned only).

c Refers to total non-tensioned tensile reinforcement.

d Refers to concrete strength at 28 days.

e Refers to modulus of rupture of concrete as measured from laboratory tests on plain concrete

specimens.

Ap 28

TABLE CB

a DETAILS OF BEAMS REPORTED BY BURNS & SIESS (54)

b c I d I e f As fcb fc Beam (in2 }

Eccentricity y Remarks (psi) (psi) (psi)

J9 1,58 8.00 510 4190 47.0 All beams had a

span of 12 1 -0 11;

JlO 1. 58 6.00 474 3590 45. 1 The reinforcing

steel had on elas -

Jll 1. 58 4.00 505 4110 46.9 ticity modulus of

30 x 106

psi.

a All beams had a width of 8". The total depth for beams J9, JlO, and Jll was 20", 16 11 and 12" respectively. All beams were centrally loaded.

b Refers to the total tensile reinforcement.

c Refers to the modulus of rupture of concrete at the time of the test.

d Refers to the concrete strength at the time of the test.

e Refers to the yield strength of the reinforcement.

Ap 29

APPENDIX D

Appendix D includes the details of the common

cases of prestress moment profiles along with

the formulas for computing camber.

Ap 30

APPENDIX D

COMMON CASES OF PRESTRESS MOMENT DIAGRAMS WITH FORMULAS FOR COMPUTING CAMBER

Prestressed Beam F 0 e Moment Diagram

Midspan Camber Due to F

0 e Moments

e l OT

(A1· )F = F eL2 /8 E .r O Cl g

0

12 E . I Cl g

(A ) _ F (e +e ) ~L2 Zj F e L2 i F - o c o _ -~ _ o o o E . I 8 6 SE .I

Cl g Cl g

Ap 31

APPENDIX E

Appendix E includes photographs of the laboratory

specimens during the various stages of the experi

mental program.

Ap 32

Figure El View of laboratory showing beams in foreground and prestressing bed containing additional beams at right.

Figure E2 Forms for beams in prestressing bed

Ap 33

Figure E3 Strain gage indicator and switching and balancing unit used with load cells to measure prestress force

Figure E4 Pres tressing bed, jacking equipment and beams stored in bed

Ap 34

Figure ES Close-up of jacking equipment, bulkheads, and grips

Figure E6 Shrinkage specimens in foreground and 7 beams ( 1 beam c ros swis e in foreground). Two additional beams in prestressing bed

Ap 35

Figure E7 Two of 4 composite beams. Strain gage points and dial gages can be seen. Strands used in relaxation tests are seen at right

Figure EB Cylinders loaded in creep racks and Whittemore gage used to measure strains of beams and shrinkage and creep specimens

Ap 36

Figure E9 View of beam Cl showing the crack pattern prior to failure

Figure E 10 View of beam C 1 after failure

Ap 37

APPENDIX F

Appendix F includes the following:

(i) A 'loss of prestress and camber' flow chart, its explanation and a typical computer output for interior girder No, 153.

(ii) A 'load-deflection' flow chart, its explanation and a typical computer output for laboratory beam Al.

Ap 38

Start

Read Input Data

{For details, see explanation of flow chart)

Write Input Data

Initialize all variables

Compute the correct ultimate creep and shrinkage coefs using

a sub•routine

Compute the elasticity modulii of beam con-crete at release and at slab casting as well as the elasticity modulus of slab concrete at

28 days

6

Ap 39

Compute the moments and the deflections (initial} due to slab and diaphram loads

Compute the initial loss of pres tress at end and center of the

beam

Determine the effective initial prestress force (after elastic

losses}

Compute the shrinkage and creep coefficients for 'time' required as well as for the ultimate conditions based on type of curing of the beam and slab concrete. Depending on the 'time' parameter, determine the loss of prestress due to shrinkage.

Ap 40

Depending on the 'time' parameter, compute the loss of prestress due to steel relaxation and due to creep of

concrete

Compute the total loss of prestres s

Compute the net initial camber due to prestress and beam dead

load

Compute the time -dependent camber due to prestress and time-dependent deflection due to beam dead load. Compute the total camber if 'time 1 parameter corresponds to a period prior to

slab casting

3

Ap 41

Write results if 1time 1 par Continue if 'time' parameter ameter corresponds corresponds to a period after 1------lperiod prior to

slab casting slab

End

C~mpute the loss of pres tress at end and center due .....- !( •

to creep depending on the type of curing of the beam

concrete. Similar values are obtained corresponding

to the 'ultimate' stage also. Compute the elastic and

creep gains due to the slab and diaphram loadings.

Compute the gains due to differential shrinkage also.

Ap 42

Compute the total loss of prestress corresponding to the 'time' parameter and the 'ulti-

mage' stage

Compute the initial camber due to prestress, the initial

deflection due to beam dead load and time -dependent camber

and deflection prior to and after slab casting due to prestress

and beam dead load respectively. Also compute the time-

dependent deflection due to slab plus diaphram loading.

Determine the deflection due to differential shrinkage and

then the total camber corresponding to the 'time' parameter

and the 'ultimate' stage.

Write results of analysis

Ap 43

EXPLANATION OF FLOW CHART FOR LOSS AND CAMBER

SL No.

1-2 1

22-50

51-96

97-98

99-115

116-120

121-134

135-139

140-190

Explanation

The read-in data includes the unit weight of beam concrete, unit weight of slab concrete, beam concrete strength at release, beam concrete strength at 28 days, slump of beam concrete, slab concrete strength at 28 days, ultimate shrinkage coefficient of slab concrete, elasticity modulus of prestressing steel, gross properties of the beam section, initial prestressing force, ultimate creep and shrinkage coefficients of beam concrete (referred to standard conditions), thickness and gross area of slab section,relative humidity, age of beam concrete at release of prestress and at slab casting, identifiers for type of curing and type of cement for beam concrete, diaphram loading and diaphram deflection, composite section properties, time parameter, and the correction factors for creep and shrinkage coefficients for the 'ultimate' stage.

Write input data.

Initialize all variables.

Compute the correct ultimate creep and shrinkage coefficients using a sub-routine.

Compute the elasticity modulii of beam concrete at release and at slab casting as well as the elasticity modulus of slab concrete at 28 days.

Compute the moments and the deflections (initial) due to slab and diaphram loads.

Compute the initial loss of prestress at end and center of the beam.

Determine the effective initial prestress force (after elastic loss es).

Compute the shrinkage and creep coefficients for 'time' required as well as for the ultimate conditions based on type of curing of the beam and slab concrete. Depending

191-209

210-213

214-225

226-237

?38-307

308-352

353-386

389-467

Ap 44

on the 'time' parameter, determine the loss of prestress due to shrinkage.

Depending on the 'time' parameter, compute the loss of prestress due to steel relaxation and due to creep of concrete.

Compute the total loss of prestress.

Compute the net initial camber due to prestress and beam dead load.·

Compute the time-dependent camber due to prestress and time -dependent deflection due to beam dead load. Compute the total camber if 'time' parameter corresponds to a period prior to slab casting.

Compute the loss of prestress at end and center due to creep depending on the type of curing of the beam concrete. Similar values are obtained corresponding to the 'ultimate' stage also. Compute the elastic and creep gains due to the slab and dis phram loadings. Compute the gains due to differential shrinkage also.

Compute the initial camber due to pres tress, the initial deflection due to beam dead load and time -dependent camber and deflection prior to and after slab casting due to prestress and beam dead load respectively. Also compute the time-dependent deflection due to slab plus diaphram loading. Determine the deflection due to differential shrinkage and then the total camber corresponding to the 'time' parameter and the 'ultimate' stage.

Write results of analysis.

This is a sub-routine to apply correction factors for the ultimate values of creep and shrinkage coefficients.

Ap 45

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Ap 46

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91 ToRIOl =O. ~4 t;L 1 A!=O.

Ap 47

95 01=0. % ~L T=O. -l7 T•Tlf'.E 9~ CALL C0 :cr 'q EC!•3),ow10•1.~•Fc~o•0,5/IOOO.

100 ESlt~=33.~~2~$l.5#FC223~*0.5/lOOO.

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121 lN=ESl/E~I 122 FSl=l=l/f1S 123 AT=llG+lT't-1.)t'~\S 124 Glt=Gt+(T~;-i.)*~5~CE~EE

125 GIC=GI+IT~-1.l*AS•EC*EC 126 C•fl/,\T 127 ClE=~I~~~~~E/Glc

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155 156 !57 15n 159 !60 lb! 162 163 164 165 166 167 !6P l~q

170 171 172 173 174 175 176 177 iB 179 \RO 161 I 62 1~3 164 185 186 187 16~ \89 190 191 1?2 193 19't 1Cj5 196 197 !SB IS~

zoo ;01 202 203 (04 20? ?06 201 709 209 210 211 212 ?13 714

Ap 48

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704 PRY=T><0.6 CT•CU"'"HY/(JO.+PPYI PRYl=(lS-111**0.6 CTTT•C~••PAYl/ltO.+PPYll TCCTl•lT+fl-7ol/(35.+T+Tl-7ol TCCT?•(Tl-7.l/13oo+Tl-7ol TCCT=ITn.nTt-TCOT21•:o.••l-6oOI ESHT =t SH1W<TCOT TCCT,=ITS-7.l/(J5.+TS~7.J rrCT4=1TOCT3-TOrT21•tO.•R(-6.0I ESHTss~rcer1.i1<i::SHlJM

1CCTS=ll·-TCOT21*10.••(-6.0I 705 er..~ r u:uc

P=fJS//l.t; TKS1=1.t1G•~5•EE/GI 1KS2=1.+~G•EC*~C/GI Tll=TN<P•TKSl 112=g•P•TKS2 ES~LEI •2SHT•"Sl•IOO,/IFSI•ll.+TIJll fS•LC! =cSHT•fSl~!OO./!FSI~ll.+Tl211 ESSTSC•iS•TSS•IOO.•ESl/IFSJ•l!.+T!211 ESSTSE•ES•rss•100.•ES!/IFSJ•l1.•T!Jll ESH·~T·~SHur•TGOTS•~SI•IOO./lF!l•!I.•Tllll•CnRTT/SCR4 !S•C~T•!S~UF•TCOTS•ESl•IOO./IFSl•(J.+11211•CO~l1/SCR4 Tl•T•24. CFPI •l.S•ALOC.IOITll T2i?•TS•?4. DF~Ssl.5•'ALCG10(T212J CF:tLLT::7.5 IFIT-!TS-Tlll706,706,707

70£> !F(T-30.170.,,700,709 708 TKK•f•0.'./30.

GC TC iiO 70q !F!T-!'0•1111•711,712 711 TK•=O.;+IT-30.l•O.!/l;.O.

cc Tr 710 712 !F!T-l:.•Jc0.ll71l,7!3,714 713 TKK=0.2+1T-l~O.J~O.O~/l620.

Gt: TC liO 7l't Tr<K=O. ?~· 110 cr.•,Jh~l

CL'l•C~'•CT•!l.-ITKK/2.011 CLCl•C!C"CT•(l.-(TKK/2.011 CUPSl•Cl.d CLC~S!•CLCl TL:=C~;+~LEl+~SHL~!+GF?l TL2 =C3C+CLC l + ~ SHLC l +l•t Kl lFl~'PI 71~t71':i,716

Ap 49

215 715 CELTO=Fr.•~COSP•SP*l44./(A,•ECl•Gll '-lh GC Tr. 12:; 217 716 cc•TI••IE 21! CRYl=tC•SP•SP/R, 719 C~Y2=1 'C-Ed''li?•HP/6. 220 CRY=CAYl-CCY2 221 OELTOJ=FO*l44,0CRY/![C!•Gll 222 725 cr•TihLE 223 Cl,=Nl<SP<SP•l.5/1000, 224 CELT~2=15.•0L~•S••SP/!GI*ECil 225 CELTf=CELTAI-DELTA2 226 f'1l=FI1'0 '221 JF(HP-0.1726,72.6,727 22P, 726 CFl=(TLl+TL2-C2E-C>Cl•0,5 2<9 GC Tr 728 230 727 CFT=TL2-C3C 231 72P CCNTI•L~ 232 CFl=OFT•FOl/100. 233 TER,11 =!-CFl+(l.-!DFl/2,0ll•CTl•CELTAI 234 TEq,22 =-CT•O·LTA2 235 Dl=C~LTA+TERMll+TE~~22

236 TER,..3l=TER~ll

237 TE•'5l=TcR"22 238 GC TG S999 239 707 ca~Tl\UE 240 lf(T-30.1730,730,731 241 730 TK;<=T•O.t/30. 2'+2 CC TC 740 243 731 IF(T-1FQ.1732t732,73~ 244 732 TKK:::O. l-t I T-JQ. l::=O. l/150. 245 GC TC 740 ?46 733 1F(T-(3.=J6Q,) )734,734,735 247 734 TKK=0.2+1T-lo0.l<-0,05/1620. 24R CC TC 740 249 735 TKK=0.25 250 740 CC~Tl~U~ 2';>1 CCf<IP=CU~;*CCRT /CfJ<:<..4 2?2 CCSP=CGS*CCl{T /C(l~i, 253 CLEest =C2~•CfTT•(J.-(TSK/2.ll

254 CLCESl =CJV•C TTT• I 1.- I TSK/2, I l 255 CL~/!Sl =C2E~CCT-CTTT):.!:(J,-(fSK+TKl<l/2.l*R~TIO

756 CLCASI =OC«!CT-CTTTl*! 1.-!TSK+TK'l/2. l*R'11C 257 CLELLT=C2E~•(Cfl~~-CTTJ)u(1.-10.25+T5Kl/2.JoR~TIC 258 CLCLLT=c;c~1c1~~-crrTl•ll.-IQ.25+TSKl/2.)*~ATIC

759 CSiLLT•C2~•1cos•-CTTTl•(l.-!0.25+T!•l/2.l•~·r10 200 CSCL;LT=C3C~lCOS::>-CTTf):i:·( 1.-t0.2S+15Kl/2. )t~bTlC 261 ~G~=O, 762 ffiCl =-[SJ~rEL~*EC*!OO./IECS•Gl*FSll 2u3 JF(IG-!)770,771,771 264 770 CTo•CU"c''·( f-(TS-TI} l**0.6/(10.+(T-TS+TI l<*0.61 265 GC Tr:' 77'0 266 77! CE=CUSS"IT-TS+Tl}H0,6/(10.+!T-TS+Til••0.61 267 775 CC•Tl\G' 26& CGS~l =~G~··CT5*RJTir 2o".} CGSCl==' Cl~Cf):;:F,~TIC ?70 CG~LLT= ~~~cuSS*RATIC*Cr~T/CCR4 271 CGCtLT= GC!t:C:lJ;·1 ~~q;,1 IC>il-CO~T/C.l.14

272 ~LECLT= G~~·CLIMS*R~TJCt:Cr~T/CrR4

273 GLCLLT= ~Cl~CU'4S*~~TlC~cnRT/CCQ4

274 lFI 11~-0 7~1 0,7J0,781

27~ 276 771 27R 27q 2RO ?Bl U2 2H3 2H4 255 266 287 28~ 289 290 291 2 ~2 293 294 295 296 297 298 29~

300 301 302 303 304 305 30b 307 308 3og 310 311 312 313 314 315 31~

317 3IR 3i9 320 321 322 323 324 325 3<6 327 •?& :120 330 ~31 3)2 133 334

Ap 50

780 yqyJ:( 1-TS+TI J/{3,.+T-TS+Tll TRYZ=(TS-2.5>/t~5.•TS-2.5) T~Y?2=1T+T!-?.511!5).+T+Tt-2.51 ClfFl•:s1ruS•TRYl-ESPUM<!TAY22-T•Y21•CORTT/5CK4 PR 1 = ~ SHUS- o SIUJ;I" I l. -TR Y2 I ~cnR TT /~C~4 GC Tl'.: 7';0

781 CGUll/U[ TRY!•!T-TS+Tll/!3•.+T-TS+Tll H·v3=!TS-7.l/!Jo.+TS-7.l TRY32•!1+Tl-7.l/!35.+T+Tf-7.I r.t ff l "'' SttUS•rn Y 1-E SP1lt1•(Tqy 32-TR Y 3 l •CCq TT /SC~4 PR!•eSHUS-oSHU~*!J.-TRY31•CCRTT/SC•4

790 CC~Tl,~E PRll•IHS!PRll•tl0.••1-6.011 C!FF•A•S!CIF•tl•l!0.*•!-6.011 CULT•E5L•~•P~!!•AG2•ESL~B/!3.•ECSI C=ESLAACDJFF~!~2~ESLAR/C3.~~cs, PGCCl =-fS!•O•YCCS•rcc•100.1rECS•C!!•FSll PGCEl ·-•sr•u•YCCS•ECe•100.1recs•c11•FS!I PR' y ?•- ~s r ••JllL r*-YCC S•£CE~ JOO .1 {EC S•C r l •FS ! I PRAYl=-ESl•GULT•VCCS•!CC•iOO./!~CS•Cll*FS!I Tll•CZ'+CLEas1+CLEISl+ESHLti+CFRl•EGE+CGSE!+•GDEl TL2•C3C+CLChS!+CLCISl+tS~LC!+rFRl+EGCl+CGSCl+PGCCl lFtJn-117758,778~,77~9

7788 Tl"LLT=CZE•ClEBSl +CSEUlT+ESHr•T+CFRULT+Fv(+GLEULT•PR•YZ TLCULT=CJC+CLCRSl +CSCULT+ES~CPT+CFRULT+E5Cl +GLCULT+PRAYl GO tr 7790

778~ CS~ULT=CLEULT CSCLLT=CLCULT GL=LLT-=CG'..:ULT GlCtl T•CGCLLT Ge rn 77se

7790 cr;r,r1r-;c~ IF(~D-0.JHOO,R00,~01

600 UELTll=F~•EC•SP•SP•!44./(8.•EC!•Gll CFT•!Tll+TL2-C2~-C3Cl•0.5 DFTLLT•ITL'ULT+TLCULTl•0.5-!C25+C3Cl•0.5 CC TC '02

801 CC" T U;Lo CRY!=EC•SP•SP/P. CRL=t~C-cE1*HP~~P/~.

CRY•DRY!-DPU CELT<l•F"•l44.•CRYl!ECl•Gll CFT•TU-OC CFILLT=TLCLLT-C1C

R02 CCI\ T t ·':t':: CL~=~l'S~*SP~l.~/1000. CEL1a2~15.eCL~'~"*SD/(G[~~cIJ CEL1t=t~LT~l-O~LT12 f'-0.i =<F J /f.:1 CF!LLT•OFTULl•FOl/!OO. CFl•~Fl•FOl/lQO. AE=ESSTS~+CFPS+(2~+C7E*(!.-(TS~/2.0JJ~CTTT ~C="SS1"C•Cf-'S+C3C+C'C<f 1.-t TSK/2.01 l•CTTI JFf•IJ-0.)~:40,~:40,~41

~40 lAVG=tu:+~()=~0.5M(C2:+C3CJcO.~ GC 1r :· '.>O

P,41 R~VC-=~C-C3C

nso CC\Tl\li' RAVCl=F•VG<FOl/!00.

335 ~~6 337

338 B9 340 341 342

343 344 345

346

347 34 P,

349 150 151 352 353 354

'355 356 357 35R 359 360 361 362 363 364 365

. 366 367 366

369 J70 HI

372 173 174

375 H6 l77

Ap 51

TE~f'Jql =-~~VCCS*Sµ*SPtl44./(8.~EC5~CJ11 TEOPll •l-A>VGl+ll.-O.~•RAVGIJOCTTTl•D[LTA! TER,41 •t-1nFl-RAVGll•ll.-lrFl+~A~Gll•0.5li•ICT-CTTTl

;i>t.OELT/\I*:-~~\TJn

TERP51 •-CTTT~OFLTA2 TfR,~l ·-1cr-CTTTl•OELTA2•RATIO Tc•P71 •-CELL TE•'~l •-Cl5*1lELLO~AT!O CI·=CELTA+T~RF3l+TE1<~4l+TER~5l+TER~fl+TER~7l+TER~'81

•+TE!'t~~l

UL 1 f\ ! = -Qt_I LT *YC c s:i:•s p,,, Si>* l '· 4 ·' ( e • * E cs O:<C 1 I ) tlLT.b2=T!::;..;~~! ULTSR•l-IGFlULT-RAVGll+ll.-IDFlULT•RAVGlJ00.5ll*ICCSP-CTTTI

~+CCLTAJ*;{.'ifl.J

ULTl1•l-IUFIULT-R>VGll+ll.-IDFlULT•RAVGll•0.5JlOICC"P-CTTTI *(.:C~LTAI*~.ATID

UL f_A4=T::r>f'J51 ULT1"•-ICCPP-CTTTl•CELTl20RIT!C UL r A6=H<nl LLTA7=-CUSS•nELL*RATIC*CCRT/CCR4 ULT AB=-CllMS•QoLL*R>T I C<•COU /CCR4 ULfA9=-<CCSP-CTTT)¥CELTA2*~hTIC ~RIT~(0,2101c2:,c3c,FO,DELTA

210 FO~r-'ATllH ,}ic., 1 :::L. LCSS tErJC) *4X,'~L. LCSS lCTR> ••x.•PRcS. FC'<CS Fr. (KJo'SI *4Xo'lN!TllL C~~PER l!NI

IFtIC-1)3533,5534,5534

'F25.2// 'Fl4.2// 'Fl4.2// •F14.2//)

5533 UL 1 •C2L r i.+CLT A2+UL T SO+IJL T A4+UL TA9+Ll TA61UL Ta l+UL TAB GC rr. 5535

5534 crnI~l" UL H~=IJLT A7 ULTll:-.t=tJLT,\5 VLTSK=LLT/11 GC TC 5>13

5535 er" !'\Uc <;Ci'79 CC~T!~.1.;::

~R.IT:::tr),llilllT 11111 FO~~Al c:~1 ,6X1'LOSS AT q~A~ :~c AT TIME= '• F6.1,2x,•cavS 1//)

h~JT~f6,9501) c2~,ESHL~l,CLE~S1,CL~~s1,cF~1,~GE,CGSEltPGCEl,TLl c;5:;1 FO.?":\T(lt-i ,3x, 1 CL. LOSS 'F25.2//

>:c4Xt 1 SH,~K LCSS 1 Fl4.2// *4(, ·c~:::::P H::~r~?E SL.O.B Cl\ ST 1 fl4.2// *4X1 •:r{;;:::o .:.r:T~Q. SL,\3 CJST 1 Fl4.2// *4X, 1 ST'=:CL ;:;.t:LAX. 'fl4.2// ~=4X, 'il. C!IN 1 Ftlt.2// ~4'( 1 1 C~"":ED Cl!~~ 1 Fl'•.2// ¥4X 1

1 G/1t·i CL;:: T•J JIFF 51-Rll'lK 1 fllt.2// »4X, 1 TCT\L Li:SS 1 Fl'1.2//I ~~lt~tt.,11112)

11112 FJ 1~WAftlH ,h<, •t_nss 'r 8~A~ ~~ctr utrr~1r~ •111 V;~lT (h, ~5011 C2E,::;SHE:1"l,CLECS1,C5€ULT,CFKL:LT1EGE,GLEULT,

~p~J'I' , lL ;·UL I iojQJT ( 1)tllll31T

lllll f:~·~ ..... T1!~· ,f;<,'LOS:. ,,r i;i:~M CTR. AT r11-•c = 1 F6.1, ?x,•CA'l'S'//) ~~IT (•1,1~011 C3C1ESHLC!,CLCes1,CLCASl,Of11,~GCl,CGSC!,

~p::;cc , YL 2 iilll (.'.,,11114)

11114 F~~~ T(lii ,6<t'LOSS hT flEAM CT~. 41 ULTl~~l~ 1 //) hRIT l~,J~Oll c1c.~S!iC~T,CLCRS1,cscuLT,oFR~LT1EGCl,0LCULTt

37d 37'1 3EO 3a~

362

Je3 3h4 385

3811 387 388

369 3qO

391 392 3?3 394 395 391> 397 398 3qq 400 401 402 403 404 405 40b 407 408 40q 410 411 412 413 "l '1 415 416 417 '•l H 4lq 420 421 422 423

Ap 52

ORAYl, TL CULT hRITFIAollll51 T

1111s Fo~~'111H ,bX, 1 M1osPAN tA~oeq AT 11~e = • F11.1. 2x,•0Avs•111 CELT ,\2=-0EL T~l WRITFCb,qs:21nELTA!,O!LTA2,TrA~31ol~RH4l,TERH5loTERH611 •TE~~11,r~~~Rl,T[~~?1,c1

9502 FO~~•TllH ,Jx,•Cu~ OU" TO PRES. •4x,·o~. o~•O LOAD OEFL. •4x,•CRP. C,H~ B~FO•E SL8 CAST •4~ 1 1 CRP C~o•• AfTE~ SL~~ CAST •4X 0

1 CRP OEFL DEFCR~ SLAB CAST •4Xo 1 CRP Offl ~FT!R SLA6 CAST •4X,'El. SL~B O~Fl *4X 0 'CkP UtFLo OUE TC SLAB ••x.•OcFL. cu: TO O!FF. SH'-K *4X,•TCIAL CEFLECTICN CR CAHB~R

WR IT: I ~, 111 l 71

'F25.2// 'f\4.2// 'fl4.2// 'fl4.2// 'Fl4.2// 1 Fl4.2// 'fl4.2// 'fl4.2// 'fl4.2// 'f14.2//)

11117 FCRPATllH .6~.·~lDSPA~ c••ecR Al ULTIMATE .,,, wRITcl6,~5'.2) DELTAl,CElTA2,ULTA2,ULTSR,ULT~4,ULTA9,

•UL U6, UL TH ,Ill T•J, Ul T wqJTCC6,225351 CALL ~XlT ~NC

sue~OUT!~E c•E~P cc~ •CN /C'-lf/CU. E S>!U. CUM ,cus .cu~ s •. cu~ s. E SHUS. TE SHU. T 1.1 s' TK l .H, •=s~~M,JO,s!,C0~4,S0~4.CORTT,C~RT

lFITS-0.Jl.1•2 l CCRl•O.

GC T~ l 2 lFllC-114,5,5 4 CORl•l.:3•TS••1-0.0951

GO H1 3 5 to•1=1.~5•Ts••1-o.11s1 3 COliTl~ll:

1 F l IC-! I b, 7, 7 6 lFITt-J.)9,9,8 qCCR2'1.

GC TC 10 8 co~2•1.1J•rr•*l-o.09s1

GC TC 10 7 lFITl-7111,12,ll

12 CC~2=1. GC. TO 10

11 cc~2~1.2s~rI*•<-o.11!1 10 CC' 11 ";'

IFCSl-3.}13113114 13 CC~:?=l.

~C TC !.5 14 C~~J=0.•2•Q.0A7RSJ

SC~3=Q.~q+Q.0407•S!

l; CC'Tl"-< !F()Kl-L,ll~,l&,17

If. CC'<4= l. 5C"4=1, r,c· rn !H

17 Cf:,14=1.!2-0.oz«TKl SC04=l.l13-0,0J2•TKI

IS CO'Tl~l;E

Ap 53

424 IF ( f--40, I U ,lg, 20 425 19 CCd'=l. 426 SCR5=1, 427 GC TC ?1 428 20 IF(l--90.)22,~~,:3

42Q 22 C~~5=1.27-0.0067*H 430 SGd~=l.40-0.0l•H

431 GC Tr 21 432 23 JF(l--100. JZ4,2S,2') 433 24 CCR5=1.27-0,Q0670H 434 SCl~S=3 .o-o.o-;*H 435 r,c Tr 21 43~ 25 cr~11•u~ 437 wR!TE(6,20ll 438 ~:.1 fO;<r.\T!ll! , 'l'NALIC DATA CCNCER'<l~G HU~!C!TY ') 439 GC Tn ~ SB8t 440 21 CCH p,u: 441 CU• =CU' C'lR2'•COi\3¥( CR4*C0'5 44 2 CU S =CLJ-'cCf1P: 2 ·~CC:l -:t.,:1CCR4*CCR. 5 443 CU~S=ClJ~CCRl~Cf1~3*C~R4*COR5 441, GUS S=C!..: ¥CC~ 1 1~c0;:,.. 3 :i::ccq4 *CC~ 5 44 5 tSliL~= i: SHU* SOR 3* S01~4::,5 C~ 5 446 cSf-US= T 'SHU 447 JF!ID-1126,27,27 44~ 26 CU~=O. 449 CUSS=O. 450 GC TC 30 451 27 CUS=O. 452 CU"5=0. 453 30 cc~.1r.'n . .:r. 454 WRITF(6,l200) 455 12·:c FIJRl"iiT(lll ,1··x, 1 c c ~ p u TE c RE s u LT s '//) 456 Al=CUi·'.~·t.:rf-'.l.l/C0~4

45 7 02 =CUS'''Ct''\ T /C0:~4 45& B3=(lJ~S*CC~f/CQR4 45q C4=Cl!SS,:=(r·~T/Cl'Q.4

460 eS=tS~Ui~~c~~TT/S~l~4

461 \\RIT'.:(6,?00) 462 200 FC"-1"1TI ili , 1X 1

1 CR.ECP A~ID SHRlf\KAGE CO(FFS I~JCLUDir\G CCRR FACTCRS' •11 ll

403 kRTT~l~12000l~l,821r3,P4,B51ESt·US 464 2000 ro•::11 t•l(lH ,·3'(, 1 lJt.T.CQP CCEFF--/J.c. 1 F23.2//

*4X, 1t1LT.C .. ~~.r:c1::r-r--s.c. 1Fl4.?// ••X,'ULT. C'<P C~EFF-- SL~ CN S.C. 'fl4.2// *4X, 1 ULT. (f.'P cn::FF-- SLe c~- ~·.c. •rt4.?// •4X, 1 ULT.::.1-'~k cc~~FF--P~~Cl\ST f'~· •fl4.?// ""4X, 1 \JLT. 51-f?_K, co:::FF---SLC fR['!v' CtiYl 'Fl2.2//)

465 E938• CL\Tl'.L" '-66 R[TL·H\ t..o 7 Ei\C

I N P U T 0 ~ T A

II c A ~ c c N C R E l

UNIT ~T, IPCFI

CC•lC, STtl. AT f!El.

CCNC. STQ. AT 28·0~Y

CCNC. SLU~P ll~CHt?SI

S L A B c c

UNIT ~T !PCFI

CCNC. STq~, !PSI I

I PS 11

!PS.fl

~ C R E T

e e A M S E C T I C N

GROSS APeA 11N0•21

GROSS ~I (IN .. 41

ECC. AT.~NO 11~1

ECC. AT CT~ Cl~I

lrT l

STEEL AREA llN••21

LEAST Cl~ CF ~E~!. 11~1

HAR?l~C U!ST, (FTI

INITIAL o~EST. FCRC~ CKl'SI

ELASTICITY ~CD~LUS (KS!I

ULl.T~~.CCR~. FACTC" FC~ cqp

lJLT.T~~.cc~R.FjCfC? FC~ SHqK

S L A H secr1n'~

E

e

Ap 54

122 .oo 4670.00

5980.00

3.00

150.00

3500.00

519.50

108512.00

6.ZO

l'e.30

86.00

4.56

s.oo

34.40

867.00

2~000.00

0.94

o.c;o

1.00

Ap 55

SLAB All E:\ ( l~ .. 21 588.00

OLM CLf TC GI AP 1-. I IN-KIPS) 776 .oo DEF. CLE TC c j,I. I l 'll 0.23

T I r. E: - 0 s p E N 0 F. N T F A c T c R s

o~ AGE >T REL lC'YI 2.00

A Ge CF "" hT Slflt1 c~srw~vs1 67.00

AV!:fL~GC REL. HL~. ( """) 10.00

c 0 N p . s !? c T . D E T A l L s

CO•P. rt Clf\**4) 331167.00

F:CC. CF DI FF. F CP.C E t I': I 13. 56

CGS GIST AT !:NC I I~ I 29.'20

CGS CIST >T CTR I I« I 21.20

RAT IC IUIC o.33

C 0 r P L T E D ~ E S U L T S

CREEP •~D SHRl~KIGE CCEFFS l~CLUOl~G CDRR FICTCRS

ULT.CRP CCEFF--•,C,

ULT.C~P.CCEfF--s.c.

ULT. CPP CCEFF-- SLL C• S.C.

ULT. c~· CCEFF-- SL• c~ r.c.

lfLT.S~~K cc:FF--~k2CjSJ fi1~

ULT. S~RK CC~FF--SLI'. FkC~ O~Yl

o.oo 1.62

o.oo 352,,EO

330.00

Ap 56

N I T I A L S T A T E

EL. LCSS lE~DI 9.01

FL. LCSS lCTRI 12.03

PMES. FrRCE PO lKIPSI

INITIAL CAPeER llNI

LCSS AT BEA~ ENC 'T Tl~[ =

EL. LOSS

SHRK LCSS

CREEP eEFORE SLAO CAST

CREEP •FTER SL~C CAST

STEEL RELAX.

El• GAii:

CRoEP GAIN

GAIN CLE TC CIFF S~RINK

TOTAL LCSS

LCSS AT BEAP ENn AT ~LTl~ATE

FL. LCSS

St<RK LC SS

CRE~P efFCRf SLA3 CAST

CREEP ~fTER SLAP. C>ST

STEH R[LAX,

EL. GA!~

CRE<oP GAIN

GA!~ OLE TC ClfF s~rJ~K

TCHL LCSS

9.01

4. 54

7.69

i.oe

6.1~

o •. oo o.oo

-0.47

28.05

9.01

4.78

7.f~

1.10

7.!:0

o.oo o.oo

-0.44

30.24

5l0~0 CAYS

(L, LCSS

S11RK LCSS

CREEP e~FCRE SL•e C•ST

CREiP AFTER SLAE COST

HEEL HLAX.

EL. GAi~

CREEP GAIN

GAi~ CLF TC CJFF s~~r~·

TOTAL LCSS

Ap 57

12.03

4.2~

10.26

J.44

6.1>

-4.20

-l.3E

-0.64

LCSS AT BEAP CTR, AT ULTIMATE

fl• LcSS

SHRK LCSS

CREEP !EFCRE SLAB CAST

CREEP AFTER SLAE CtST

STEEL f!ELAX,

EL. G.01~1

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TCT•L LCSS

P!DSP•N CA~BER ~T T [ t1 t

CBR c IJ ~ TC pq::s.

AP. CEA!'. LCtC CEFL.

CRP., Ct--P.~ QEFCRt $LB CAST

CRP c fJ i' P.. •FTER SL .:it\ C t..S T

CRP C EFL O'OFCRE SL 'fl CAST

CRP CEFL AFT[ R SL A·~ C~ST

EL. SLH' DEFL

CRP CtFL. cu~ TC SL 1\1)

560.0

12.0;

". 5 2

10.26

2.21

7.50

-4.20

-1.68

-0.Gl

30.03

CAYS

3. P.7

-l.64

2.3>

0.46

-1. 4 <;

-0.24

-2.21

-o. 73

Ap 58

DEFL, cu; TO C!FF, $11~)( -0.20

TCT6L CEFLC:CT!C~ O• cws:R 0.21

~ICSPAN CA HEH •T ~LTIMATE

CBR c~~ TC P~E·s • 3.87

ll~. CEJ.C LCAC C~Fl. -1.64

CRP. C~P.~ ~EFCR( SL£ CAST 2,3q

CRP c•eR. AFTER SL/oC CAST o. 71

CRP CEFL HFCRt SLAB CAST -1.~q

CRP CEFL AFT::R SLAe CAST -0.38

EL, Sl~B CEFL -2.21

CRP CE fl, cu~ TC SLAB -o.eq

DEFL, cu~ TO CJ FF, SH~K -o.1q

TOTAL CEFLECT!C~ CR CH'e"R 0 •. 11

Ap 59

FLOW CHART FOR LOAD-DEFLECTION STUDIES

Read input data (for details, see explanation

of flow chart)

Write Input Data

DO Cl I = l, 70

IF (Load. GT. Ultimate Load)

Compute effective moment of inertia

and then the deflection under applied

load

I

I I I I I I I I I I

"'

Ap 60

Write Results of Analysis

r I r I I

-- - ------ -- ___ .J

Ap 61

EXPLANATION OF FLOW CHART FOR LOAD DEFLECTION STUDIES

SL No.

1-5

5-7

8-13

14-31

Explanation

The read-in data includes the beam dead load, effective prestress force at the time of test, concrete modulus of rupture, gross sectional properties of the beam, the concrete strength at the time of test, the ultimate load of the beam, the cracking load of the beam and the cracked moment of inertia.

Compute the maximum dead load moment of the beam and the cracking moment of the beam.

Write pertinent information from the read-in data.

Compute the deflection under applied load and print the results.

Ap 62

$JOO 1 KRIPP l DI McNS!tJN DllOOI 2 READl5,4441W,FT,FCB,AG,Tl,ET,YT,EC,AS,8,Dl,Al,Bl 1 SP,FC 3 444 FORMAT15Fl5.51 °' REAOl5o440IPULT1PCR,TCR 5 44~ FORMATl3Fl0.21 6 D~CL•1.s•:i•SP•SP 7 CRM•F T4 ET+ ( Tl •FT /IA G* YT I l+FCA•TI /YT 8 WR[TE(o,1011 9 101 FORMAf(lHll

io WR!TE(o,lnlPULT,PC~,TCR U 102 FORMATllH ol5Xo'ULTl~•TE LOAD IN KIPS 1 F26.5//

•16X,•C~ACKING L3AO IN KIPS 1 Fl5.5// *16X,•CRACKEO MOMENT OF INERTIA IN INCHES**4 1 Fl5.5// •I

12 WRITE(6,l061 13 106 FORMATllH ,5x, 1 LOAD (KIPS1•,1cx,•EFF MI <IN••41',1CX, 1 DEFLECTIDN (

*INCHESl'I 14 009991•1,TO 15 TY•I 16 P•TY/3. 1l IFIP.GT.PULTlGO TO 9999 18 TMCL•33.•P+OMCL 19 IFITMCL-CRMll0,10 1'.i.l 20 10 TEF=T I 21 GO TO 12 22 ll RAT=CRM/TMCL 23 TEF•Tl~(RAT .. 31+TCR•(l,-IRAT**31 I 24 12 CONTINUE 25 0( 11 =P*Al •1728 _. ( B. *Al*Al+l2 ,•Al~ Bl+ 3, • Bl*B l l /148. •EC*TEF l 26 WRITE(o,831P,TEF 1 DI II 27 88 FORMAT(lH ,/,7X,Fo.2,1sx,Fe.2,1ox,Fe,41 28 9~9 CONTl~UE 29 9999 CONTINUE 30 CALL EXIT 31 END

SENTRY

Ap 63

UL TIMhTE LO An JN Kl PS 8.07900

CRACKING LOAD IN KIPS 3, 569C'C

CRACKED MC ME NT OF lNEP.TlA JN INCHE5**4 33. ll2CJ

LOAD !KIPS) EFF Ml ( lN*•4l DEFLECTION (INCHES I

0.33 256 .oo 0.0317

o.67 256 .oo 0.0635

1.00 256.00 0.0952

1.33 256 .oo 0.1270

1.67 256.00 0.1567

2.00 256.00 0.1904

2.33 256.00 0.2222

2.67 256.00 o. 2 539

3,00 256.00 o. 2 85 7

3,33 256 .oo o. 3174

3.b7 240.4~ o. 3 717

4.00 197.01 0.4949

4.33 164.89 0.6406

4.67 1.40.64 0.8088

5.00 121.99 0.9991

5. 33 107.42 1.2102

5.67 95.87 1.4409

6.00 86.59 1.689]

6. 33 79.05 1. 952 8

6.67 72 ,87 2. 2 300

1.00 6 7. 75 2. 51 €6

1 .. 33 63 .4 7 2.0163

7.67 59.87 3. 1215

8.oo 56.82 3. 4323

Loss of Prestress, Camber, and Deflection of Non-Composite ...publications.iowa.gov/21548/1/IADOT_HR_137_Loss... · of Noncomposite and Composite Structures Using Different ... cretes

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